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The analysis of some properties for the equilibria of switched dynamic systems is addressed. In particular, the geometric properties of the equilibrium region in state space and the algebraic properties of the equations defining it are studied. Based on fundamental results from algebraic geometry the equilibria properties of switched dynamic systems is analyzed. This alternative approach allows to obtain information about the set of equilibrium points without explicitly computing it. This study is developed for three different formulations of switched dynamic systems, revealing some interesting algebraic and geometric relations in their corresponding equilibria. Some examples, including the case of a power converter, are presented for illustration purposes.

As capacitor voltages are necessary for the three-cell DC-DC chopper control, the estimation of such voltages by an observer is attractive solution in terms of cost. However, due to the hybrid behavior of this structure, the capacitor voltages may be partially or even not observable for a given switching configuration. In other words, the observability matrix associated to the capacitor voltages never has a full rank. In order to make the observer conceivable, this paper proposes a new design by establishing sufficient conditions under which the capacitor voltages can be reconstructed within appropriate specific switching sequence and not necessarily instantly. An interconnected high gain observer is then designed for the considered structure. Furthermore, the analysis of convergence is given according to specific switching sequence, and the bounds of the observer tuning parameters are numerically verified. A significant load current benchmark including a three-cell chopper is used as prototype to show the performance and the effectiveness of the proposed observer.