Energy storage systems (ESS) are indispensable in daily life and have two types that can offer high energy and high power density. Hybrid energy storage systems (HESS) are obtained by combining two or more energy storage units to benefit both types. Energy management systems (EMS) are essential in ensuring HESS's reliability, high performance, and efficiency. One of the most critical parameters for EMS is the battery state of health (SoH). Continuous monitoring of the SoH provides essential information regarding the system's status, detects unusual performance degradations and enables planned maintenance, prevents system failures, helps keep efficiency at a consistently high level, and helps ensure energy security by reducing downtime. The SoH parameter depends on parameters such as Depth of Discharge (DoD), charge and discharge rate (C-Rate), and temperature. Optimal values of these parameters directly affect the lifetime and operating performance of the battery. The proposed Adaptive Energy Management System (AEMS) uses the SoH parameter of the battery as the control input. It provides optimal control by dynamically updating the C-Rate and DoD parameters. In addition, the supercapacitor integrated into the system with filter-based power separation prevents deep discharge of the batteries. Under the proposed AEMS control, HESS has been observed to generate 6.31% more energy than a system relying solely on batteries. This beneficial relationship between supercapacitors and batteries efficiently managed by AEMS opens new possibilities for advanced energy management in applications ranging from electric vehicles to renewable energy storage systems.