# Hysteresis-Based Current Control Approach by Means of the Theory of Switched Systems
## 2 Theoretical Background and Proposed Control Law  
### General System Description of Switched Systems  
In general, a switched system is described by  
\[\underline{x}(t)=\underline{f}_{s(t)}(\underline{x}(t),\underline{u}(t)) \tag{1}\]  
where \(\underline{x}(t)\in\mathbb{R}^{n}\) and \(\underline{u}(t)\in\mathbb{R}^{n}\) are the state space and input vector, respectively. The function \(s(t)\) represents a piece-wise defined switching sequence with values specified by the set of all different subsystems \(\mathcal{S}=\{0,1,2\ldots\}\). The time when \(s(t)\) changes its value is named switching time or switching event. The dynamic behavior of a switched system is determined, on the one hand, by the dynamics of the individual subsystems and, on the other hand, by the sequence of switching among these subsystems. There are many well-known methods to analyze the stability in the sense of Lyapunov of each subsystem [14; 15; 16; 17; 19]. However, even if all subsystems have the same stable equilibrium, it may be an unstable equilibrium of the switched system; see Figure 1. Moreover, on the contrary, a switched system might have a stable equilibrium, even if it is an unstable equilibrium for some or all subsystems; see Figure 2. In both cases, the switching sequence is responsible for the stability of the equilibrium. Figure 1: Switching between stable subsystems. The Lyapunov stability of switched systems was already discussed in literature and is a proven tool for stability analysis [14; 15; 16; 17]. However, all methods require that the equilibria to be analyzed are in the origin, which forces the designer to perform a state transformation in advance. This process is usually even more difficult in practice, as it is often not desired to guarantee the convergence of the system trajectory \(\underline{x}(t)\) to a constant equilibrium point, but rather to a subspace around a time-varying reference trajectory \(\underline{x}_{\text{ref}}(t)\). For example, this is also the case in converter applications. In the following, this convergence is analyzed for linear switched systems of the form  
\[\underline{\dot{x}}(t)=\underline{A}\left(\underline{x}(t)-\underline{x}_{ \text{eq},s}\right)+\underline{B}\,\underline{u}(t), \tag{2}\]  
with unique stable equilibria \(\underline{x}_{\text{eq},s}\) for all \(s\in\mathcal{S}=\{0,1,2\ldots\}\). If, as in the case of switching converters, the plant's input is known in advance, both the state trajectory \(x(t)\) and the reference trajectory \(\underline{x}_{\text{ref}}(t)\) can be shifted by \(\underline{A}^{-1}\underline{B}\,\underline{u}(t)\), i.e.,  
\[\underline{\underline{x}}(t) =\underline{x}(t)+\underline{A}^{-1}\underline{B}\,\underline{u }(t), \tag{3}\] \[\underline{\underline{x}}_{\text{ref}}(t) =\underline{x}_{\text{ref}}(t)+\underline{A}^{-1}\underline{B}\, \underline{u}(t). \tag{4}\]  
Since both \(\underline{x}(t)\) and \(\underline{x}_{\text{ref}}(t)\) are shifted by \(\underline{A}^{-1}\underline{B}\,\underline{u}(t)\), the convergence of \(\underline{\underline{x}}(t)\) to a subspace \(\mathcal{R}^{n}_{\delta}\) around \(\underline{\underline{x}}_{\text{ref}}(t)\) deduces the convergence of \(\underline{x}(t)\) to an equivalent subspace around \(\underline{x}_{\text{ref}}(t)\). The subspace \(\mathcal{R}^{n}_{\delta}\) is bounded by the hysteresis threshold \(\delta\) for the weighted distance between \(\underline{\underline{x}}(t)\) and \(\underline{\underline{x}}_{\text{ref}}(t)\) and is defined by  
\[\mathcal{R}^{n}_{\delta}=\left\{\mathbb{R}^{n}\Big{|}(\underline{\underline{ x}}(t)-\underline{\underline{x}}_{\text{ref}}(t))^{T}\underline{M}(\underline{ \underline{x}}(t)-\underline{\underline{x}}_{\text{ref}}(t))<\delta\right\} \tag{5}\]  
where the weighting matrix \(\underline{M}\) is positive definite. In summary, it can be supposed that there exists a switching control law that guarantees the convergence of the shifted state trajectory \(\underline{\underline{x}}(t)\) to \(\mathcal{R}^{n}_{\delta}\), if:  
1.