DescriptionThis thesis considers the design of observers for systems with slow and fast modes. The existing design method were able to achieve that goal with O( ) accuracy. Where is a small positive parameter that indicates seperation of system state space variables into slow and fast. We design independent slow and fast reduced-order observer with exact accuracy, and place their closed-loop eigenvalues exactly at the desired location. Furthermore, we apply the two stage method to design the full-order observer. The two stage method results in full-state feedback control for each subsystems which makes the subsystems asymptotic stable. Lastly, the crucial theme of the online savings is the fact that the design allows complete time-scale seperation for both the observer and controller through the complete and exact decomposition into slow and fast time scales. The above method reduces both o -line and on-line computations. In this thesis, we demonstrate the e ectiveness of the two stage methods through theoretical and simulation results.