Modern passenger vehicles are equipped with an increasing number of actuators that may be used to actively control the lateral and longitudinal dynamics of the vehicle. During limit-handling situations, proper coordination of all the available actuators by the vehicle stability control (VSC) can lead to improved overall control authority, which in turn may lead to improved handling performance and decreased intrusiveness to the driver. The difficulty, however, is coordinating the available actuators, given that the addition of actuators typically leads to the vehicle becoming over-actuated. The state-of-the-art method of solving the over-actuation is optimal control allocation, often referred to as Global Chassis Control (GCC). A drawback of most formulations of GCC proposed in literature is the prohibitive computational burden associated with the optimization. To obtain a real-time feasible GCC algorithm, the hybrid steepest descent optimization method is applied to the control allocation problem in this work. The optimization problem is set up to minimize tracking errors of virtual control inputs whilst minimizing actuator effort. A high level controller is designed that produces these virtual control input targets, which are a yaw moment used to stabilize the vehicle, and longitudinal force, to represent the acceleration intent of the driver. Using online linearization of a nonlinear vehicle model the yaw moment and yaw moment effectiveness of the available actuators is estimated and used to update the optimization problem. Furthermore, improvements are suggested to the hybrid steepest descent method to accelerate convergence and reduce or eliminate chatter at constraint boundaries by dynamically scaling the constraints. The optimal control allocator is shown, using a validated simulation model, to produce improved allocation performance when compared to a simpler control allocation method that is similar to the industry-standard. Furthermore, the proposed algorithm was converted to C-code and implemented on one of the on-board ECUs of a Tesla Model S, demonstrating real-time feasibility. Experimental results for an aggressive single lane-change using this implementation show the algorithm provides good performance compared to an industry standard brake-based stability control system.