# Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems
## 2 Theoretical Background and Proposed Control Law
### Selection of the Quality Functions  
With regard to the [PERSON] theory [20, 21, 22], it is suggested to specify the quality functions as  
\[h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))=\left( \underline{\tilde{x}}(t)-\underline{x}_{\text{eq},s}\right)^{T}\underline{ \underline{M}}_{s}(t)\left(\underline{\tilde{x}}(t)-\underline{x}_{\text{eq},s}\right)=\underline{\tilde{x}}_{\text{eq},s}^{T}(t)\underline{\underline{M }}_{s}(t)\,\underline{\tilde{x}}_{\text{eq},s}(t). \tag{15}\]  
In this context, the choice of the time-varying, positive definite matrices \(\underline{\underline{M}}_{s}(t)\) for all \(s\in\mathcal{S}\) represents a degree of freedom, allowing to ensure the convergence of \(\overline{\tilde{x}}(t)\) into the set \(\mathcal{R}^{n}_{\delta}\) and thus the convergence of \(\underline{x}(t)\) into the set \(\mathcal{R}^{n}_{\delta}\). Figure 4 illustrates the exemplary quality functions \(h_{s}(\underline{\tilde{x}}(t)_{0},\underline{\tilde{x}}_{\text{ref}}(t_{0}))\) for the above example system and some \(t_{0}>0\), where the matrices \(\underline{\underline{M}}_{s}(t)\) were chosen in a way that \(h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))=\hat{h}\) for all \(s\in\mathcal{S}\) with \(\underline{\tilde{x}}_{\text{ref}}(t)=\underline{\tilde{x}}(t)\) and \(\underline{\tilde{x}}_{\text{ref}}(t_{0})\), which is marked as red dot in Figure 4. Consequently, all conditions in (11), (12), (13), and (14) are fulfilled. It can be imagined that the choice of each \(\underline{\underline{M}}_{s}(t)\) is crucial for convergence. For instance, Figure 5 exemplifies an unfavorable and a well-designed parameterization. In both Figures 4 and 5, the color gradient reflects the value of \(h_{s}(\underline{\tilde{x}}(t_{0}),\underline{\tilde{x}}_{\text{ref}}(t_{0}))\), with lighter colors representing larger values. If the switching sequence is created by (10), subsystem 0 or 1 is activated depending on the initial value \(\underline{\tilde{x}}(0)\), marked as circles in Figure 5. As a decrease in \(h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))\) is enforced by (14), when subsystem \(s\) is active, the shifted state trajectory \(\underline{\tilde{x}}(t)\) moves in the direction of the arrows until another \(h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))\) exceeds the value of the quality function of the currently active subsystem, marked as black lines in Figure 5. From this point on, the shifted state trajectory \(\underline{\tilde{x}}(t)\) moves along this intersection line in the direction of decreasing \(h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))\), and the system only switches between the subsystems whose quality functions \(h_{s}(\underline{\tilde{x}}(t),\underline{\tilde{x}}_{\text{ref}}(t))\) coincide. The switching behavior along the intersection line is similar to that of a system in sliding mode. Sliding along the intersection line is interrupted only by reaching another intersection line. In Figure 5, this corresponds to the point were all \(h_{s}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t))\) are equal. At this point, the system switches between all subsystems, and the state trajectory remains in an area around this point. The size of the area represents a design parameter and depends on the maximum allowed switching frequency and the dynamics of the subsystems. In Figure 5a, the unfavorable parameterization of \(\underline{\underline{M}}(t)\) leads to an intersection point unequal to \(\underline{\underline{x}}_{\text{ref}}(t)\). Only if condition (13) is satisfied, i.e., all matrices \(\underline{\underline{M}}(t)\) are chosen, such that all \(h_{s}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t))\) intersect in \(\underline{\underline{x}}_{\text{ref}}(t)\), the convergence to \(\mathcal{R}^{n}_{\delta}\) is given; see Figure 5b.