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Hybrid systems combine both discrete and continuous state dynamics. Power electronic inverters are inherently hybrid systems: they are controlled via discrete-valued switching inputs which determine the evolution of the continuous-valued current and voltage state dynamics. Hybrid systems analysis could prove increasingly useful as large numbers of renewable energy sources are incorporated to the grid with inverters as their interface. In this work, we explore a hybrid systems approach for the stability analysis of power and power electronic systems. We provide an analytical proof showing that the use of a hybrid model for the half-bridge inverter allows the derivation of a control law that drives the system states to desired sinusoidal voltage and current references. We derive an analytical expression for a global Lyapunov function for the dynamical system in terms of the system parameters, which proves uniform, global, and asymptotic stability of the origin in error coordinates. Moreover, we demonstrate robustness to parameter changes through this Lyapunov function. We validate these results via simulation. Finally, we show empirically the incorporation of droop control with this hybrid systems approach. In the low-inertia grid community, the juxtaposition of droop control with the hybrid switching control can be considered a grid-forming control strategy using a switched inverter model.