With the widespread application of electrification technologies and the increasing number of advanced electrical equipment onboard, aircraft power systems (APS) are required to possess high-power energy storage and pulse discharge capabilities, presenting significant challenges to the robustness of APS. Based on the generator-energy storage system, the airborne hybrid energy system (AHES) combines the energy and power complementarity of Li-battery and supercapacitor to smooth the dynamic load power fluctuations. This paper introduces a hierarchical control framework that provides decoupled control objectives for different levels of AHES. To leverage the advantages of hybrid energy storage systems (HESS), an adaptive droop control method is proposed as an intermediate-level coordinating control to enable the autonomous operation of HESS. At the upper-level of the proposed hierarchical control, an energy management strategy based on model predictive control is implemented. This strategy has the capability to accurately predict high-slope pulse loads in the future system in real-time. Simulation results demonstrate that the hierarchical control, when facing pulse loads, achieves optimal energy allocation while ensuring system power distribution accuracy.