This paper presents a method for estimating unknown input disturbances in linear dynamic systems, formulated as an optimal output tracking problem and based on an observer approach. By introducing a virtual control input and minimizing a quadratic cost functional, an observer structure is developed that is capable of reconstructing faults affecting both the system dynamics and sensor measurements. The solution is obtained without applying model reduction or strict structural simplifications. A necessary and sufficient condition for identifiability is established, ensuring accurate estimation. The approach is validated through simulation using an electric drive example, demonstrating fast convergence and high accuracy in fault signal reconstruction. The simulation results confirm the practical applicability of the proposed method due to its high precision and convergence speed.