This paper presents a comparative study of Backstepping controllers based on Barrier Lyapunov Function (BLF) and Quadratic Lyapunov Function (QLF) for stabilising the vertical displacement of a quarter car active suspension system. Hard constraints of the suspension system such as suspension travel, dynamic tyre load and actuator saturation are maintained in presence of uncertain sprung mass. Two different case studies have been conducted for stabilization of vertical displacement of vehicle (a) using a special reference trajectory and (b) without using any special trajectory and maintaining a zero reference. The gains of controllers have been optimised for both the studies using Genetic Algorithm. Contrary to an earlier reported work [7], which used un-optimized random gain values for the controllers, it has been shown that QLF approach can provide better results than BLF approach when the gains are optimized.