# 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
The value \(\hat{h}\) of the quality functions at the intersection point is not important for convergence and is related to the choice of each individual \(\underline{\underline{M}}(t)\). In this publication, it is proposed to define \(\underline{\underline{M}}(t)\) by  
\[\underline{\underline{M}}(t)=\frac{\underline{\underline{M}}}{\left( \underline{\underline{x}}_{\text{ref}}(t)-\underline{x}_{\text{eq},s} \right)^{T}\underline{\underline{M}}\left(\underline{\underline{x}}_{\text{ ref}}(t)-\underline{x}_{\text{eq},s}\right)}=\frac{\underline{\underline{M}}}{ \underline{\underline{\xi}}_{\text{ref},s}^{T}(t)\underline{\underline{M}} \underline{\underline{x}}_{\text{ref},s}(t)},\quad\forall s\in\mathcal{S} \tag{16}\]  
with the weighting matrix \(\underline{\underline{M}}\) from (5). As \(\underline{\underline{x}}_{\text{ref},s}^{T}(t)\underline{\underline{M}} \underline{\underline{x}}_{\text{ref},s}(t)>0,\forall\underline{\underline{x}}_ {\text{ref}}(t)\
eq\underline{\underline{x}}_{\text{eq},s},\forall t\) per definition (see (6) and (7)), the time-varying matrices \(\underline{\underline{M}}(t)\) are positive definite and \(h_{s}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t))\) fulfills the conditions (11) and (12). With this choice of \(\underline{\underline{M}}(t)\), the quality functions \(h_{s}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t))\) become the ratio of the weighted distance between \(\underline{x}_{\text{eq},s}\) and \(\underline{\underline{x}}_{\text{ref}}(t)\) and \(\underline{\underline{x}}(t)\), respectively, which are all equal if \(\underline{\underline{x}}(t)=\underline{\underline{x}}_{\text{ref}}(t)\). If a switched system with only one subsystem is considered, the condition \(h_{0}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t) )=\hat{h}\) is given for an infinite number of combinations of \(\underline{\underline{x}}_{\text{ref}}(t)\) and \(\underline{\underline{x}}(t)\). More precisely, the condition is fulfilled as long as \(\underline{\underline{x}}_{\text{ref}}(t)\) and \(\underline{\underline{x}}(t)\) are on the same orbit around \(\underline{\underline{x}}_{\text{eq},s}\), illustrated as blue curve around \(\underline{\underline{x}}_{\text{eq},0}\) in Figure 6 for some \(t_{0}>0\). Introducing a second subsystem, the set of solutions reduces to two possible equilibria, specified by the two intersections of the orbits around the two system equilibria \(\underline{\underline{x}}_{\text{eq},0}\) and \(\underline{\underline{x}}_{\text{eq},1}\), marked as red dots in Figure 6. Finally, again considering the exemplary system with three subsystems shown in Figure 4 or Figure 5b, the quality functions \(h_{s}(\underline{\underline{x}}(t),\underline{\underline{x}}_{\text{ref}}(t))\) are all equal to \(\hat{h}\) only at \(\underline{\underline{x}}_{\text{ref}}(t)=\underline{\underline{x}}(t)\), i.e., \(\underline{\underline{x}}_{\text{ref}}(t)=\underline{\underline{x}}(t)\). Hence, defining the matrices \(\underline{\underline{M}}(t)\) as suggested in (16), condition (13) is also satisfied for all \(s\in\mathcal{S}\).