The optimal swing up of a single inverted pendulum (SIP) on a cart is presented, where the indirect method is used to obtain the feedforward command in the swing up mode and the LQR controller beside the high gain observer is used to stabilize the system in the stabilizing mode. After deriving the optimality conditions for a dynamic system in general formulation, the necessary conditions for optimality are derived for a SIP using the fundamental theorem of calculus of variations (FTCV) which leads to a two point boundary value problem (TPBVP). This problem is solved for obtaining the optimal values of states and control. To demonstrate the applicability of proposed method, after simulation study, a single pendulum setup is constructed and experimental realization is presented. In order to complete the swing up maneuver, LQR controller is designed to stabilize the system, and the high gain observer is applied to estimate the link angular velocity. Finally, a comparison between experimental and simulation results is presented and the efficiency of proposed method to reduce the used effort is illustrated.
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