For a nanodisk, magnetic vortex characterized by a curling magnetization is an energetically stable state. The magnetization in the center of the magnetic vortex is directed upward or downward, namely, the vortex core polarity p=+1 or p=-1 refers to up or down, respectively. The curling direction of magnetization, namely, the vortex chirality, is either counter-clockwise or clockwise. Thus, different combinations of chirality and polarity in a vortex structure demonstrate four stable magnetic states, which can be used to design a multibit memory cell. Such a multibit memory application requires the independent controlling of both the vortex chirality and vortex polarity, which has received considerable attention recently. Switching the vortex polarity has been achieved by using either a magnetic field or a current. The vortex chirality can be controlled by introducing asymmetric geometry of nanodisks. In this article, by using micromagnetic simulations, we present an effective method to simultaneously control the vortex chirality and polarity in a spin valve structure, in which the fixed spin polarizer layer is magnetized in the film plane when the free layer has a magnetic vortex configuration. The free layer is designed into a ladder shape with the right part being thicker than the left part. Our simulations indicate that a combination of desirable vortex chirality and polarity can be easily controlled by a Gaussian current pulse with proper strength and pulse duration through the spin-transfer torque effect. The insight into physical mechanism of the controllable vortex is demonstrated by a series of snapshots. If the magnetic moment of the free layer is saturated in the direction of 0θ θ is the angle between the magnetization and+x axis, the vortex with the counter-clockwise chirality will be generated after the pulse. In contrast, if the free layer magnetization is saturated along the direction πθ <2π, after the pulse, the vortex will have the clockwise chirality. The core polarity of the remanent vortex state is determined by the sign of the magnetic charges which are formed in the step-side of nanodisk during the current pulse.