# Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems
## 4 Simulative Verification
### Simulation Set Up  
Figure 11 shows the simulation block diagram implemented with the values from Table 2.  
The utilized switching law \(f_{\max}(h_{s}(\underline{\xi},\underline{\xi}_{\text{ref}}))\) is defined by  
\[f_{\max}(h_{s}(\underline{\xi},\underline{\xi}_{\text{ref}}))=\left\{\begin{array} []{ccc}\arg\left(\max_{s\in\mathcal{S}}(h_{s}(\underline{\xi}(t),\underline{ \xi}_{\text{ref}}(t)))\right),&\forall&t>t_{\delta,1}\\ \arg\left(\max_{s\in\mathcal{S}}(h_{s}(\underline{\xi}(t),\underline{\xi}_{ \text{ref}}(t)))\right),&\forall&t\leq t_{\delta,1}\end{array}\right. \tag{60}\]  
where \(t_{\delta,1}\) denotes the first time \(\underline{x}(t)\) moves into \(\mathcal{R}_{\delta}^{2}\). The function \(\max_{s\in\mathcal{S}}(\cdot)\) represents a \(\max(\cdot)\)-function extended by a hysteresis with the hysteresis threshold \(\delta_{\text{hyst}}\). Thus, a change of the output only takes place when the previous maximum value is exceeded by \(\delta_{\text{hyst}}\). This prevents high frequency switching along the intersection lines. Both hysteresis thresholds where heuristically set such that the switching frequency \(f_{SW,c,b,a}\) of the phases are in the common frequency range of a VSC from 40 Hz to 40 kHz [6]. Furthermore, the function \(f(s_{\max},s_{\text{old}})\) with two outputs \(s_{\text{opt}}\) and \(\underline{x}_{\text{eq},s_{\text{opt}}}\) is utilized to distinguish between the two vectors \(\underline{p}=\left[\begin{array}{ccc}0&0&0\end{array}\right]^{T}\) and \(\underline{p}=\left[\begin{array}{ccc}1&1&1\end{array}\right]^{T}\) and generate the corresponding  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**System Constant** & **Symbol** & **Value** & **Unit** \\ \hline normalized DC Link voltage & \(u_{\text{DC}}^{\prime}\) & 2.273 & \(-\) \\ number of pole pair & \(z_{p}\) & 2 & \(-\) \\ nominal speed & \(\omega_{\text{nom}}\) & \(3.142\times 10^{2}\) & \(\frac{1}{5}\) \\ flux leakage & \(\Phi_{F}\) & 0.587 & \(V_{S}\) \\ normalized stator resistance & \(R^{\prime}\) & \(0.334\times 10^{-2}\) & \(-\) \\ normalized stator inductance & \(L^{\prime}\) & \(9.155\times 10^{-5}\) & \(s\) \\ normalized stator time constant & \(T_{\text{s}}\) & \(2.744\times 10^{-2}\) & \(s\) \\ EMF constant & \(c_{\text{EMF}}\) & \(2.957\times 10^{2}\) & \(-\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: System constants and exemplary normalized values for simulative verification.  
Figure 11: Implemented simulation set-up.  
subsystem equilibrium \(\underline{x}_{\text{eq},s_{\text{opt}}}\). Depending on the previous subsystem \(s_{\text{old}}\), \(s_{\text{opt}}\) is defined by  
\[s_{\text{opt}}=\left\{\begin{array}{ccc}s_{\text{max}},&\forall&s_{\text{max} }\
eq 0\\ 0,&\forall&s_{\text{max}}=0\wedge(s_{\text{old}}=0\lor s_{\text{old}}=1\lor s_ {\text{old}}=2\lor s_{\text{old}}=4)\\ 7,&\forall&s_{\text{max}}=0\wedge(s_{\text{old}}=3\lor s_{\text{old}}=5\lor s _{\text{old}}=6\lor s_{\text{old}}=7)\end{array}\right.. \tag{61}\]  
The subsystem equilibrium \(\underline  
The results of the other simulations are shown in Figure 13. In Figure 13a, the state trajectories for all initial values are given in the state space. In addition, Figure 13b shows the trajectories of the state variables \(i^{\prime}_{\alpha}\) and \(i^{\prime}_{\beta}\) over time \(t\) with one representative initial value \((k=11)\).  
As can be verified, the hysteresis current controller is able to guarantee the convergence of the state trajectories \(\underline{x}(t)\) into \(\mathcal{R}_{\delta}^{2}\) regardless of the initial values \(\underline{x}_{0}\); see Figure 13a. In addition, the fast transient response typical for the hysteresis processes is shown. The state trajectory enters \(\mathcal{R}_{\delta}^{2}\) at \(t_{\delta,1}=0.05\,\mathrm{ms}\); see Figure 13b.