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This paper explains a suspension system of a vehicle which comprises of spring mass and damper.The main objective is to design a proper controller that can damp oscillations in order to provide safety, ease and comfort to passengers of vehicle during uneven road surface, bumps and small patches. To perform the desired task state space model of a quarter wheel vehicle suspension system is derived. The response of the system is simulated in MATLAB/SIMULINK. The open loop response showed that the derived model is stable. Different types of controllers are designed i.e. Linear Quadratic Regulator (LQR) and Observer based controller to analyze the system in closed loop. Both the controllers showed improved performance for different road profiles. It is observed and shown that observer based controller has better response than Linear Quadratic Regulator (LQR) controller. Simulink models for Linear Quadratic Regulator (LQR) and observer based controller are designed. Besides, different types of realization techniques (minimal realization, balanced realization, modal realization, observer canonical realization) are compared for minimum fragility in controller implementation. The difference among the different realization controllers has been analyzed in detail for rounding off error or truncation error and an optimal non fragile controller design has been presented. Different disturbances were imposed upon the simulated model. All the results are analyzed in open and closed loops. The closed loop response showed that the oscillations were damped quickly and the desire results were achieved successfully.