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Unified swing up and upright position stabilizing controllers for inverted-pendulum on a cart

The challenging problem of stabilizing an inverted-pendulum on a cart system at the upright position via a nonlinear state feedback controller is analyzed. The proposed controller is designed in a manner that can guarantee local asymptotic stability for the up position equilibrium and instability for the down one. As shown in the paper by a stability analysis and verified numerically, a wide range for such a controller gains can be easily determined. Simultaneously, a systematic qualitative study of the system motion provides the required features the input has to fulfill in order to act in a unified form: initially with a series of quick cart movements and pendulum swings which may result in getting the pendulum stick away from the unstable down position, till it can be attracted to the upright stable equilibrium. Thus, a practically parameter-free and robust design is proposed. The overall scheme is examined by detailed simulations. The response of the system indicate an excellent performance with a fast swing up period following by a convergence to the stable up position with the desired displacement; during transient, small amplitude oscillations and limited overshoots are observed.