When paralleled inverters feed a common load, it is required that the load is shared according to their power ratings. In this paper, the robust droop controller (RDC) proposed in the literature for achieving accurate proportional power sharing for paralleled inverters is implemented in a way to ensure a bounded closed-loop performance. Using non-linear Lyapunov methods, it is shown that the controller structure permits the control input to stay within a predefined range. While maintaining the main theory of the RDC, the proposed bounded droop controller (BDC) is proven to guarantee the stability of the closed-loop system in the sense of boundedness for the general load case given in the generalized dissipative Hamiltonian form, which can describe both linear and non-linear load dynamics. Extended simulation results for two single-phase inverters operated in parallel are presented to verify the effectiveness of the BDC for both a linear and a non-linear load case scenario.