Nonlinear optimal control of the planar inverted pendulum

The control and stabilization problem of the 4-DOF planar inverted pendulum is nontrivial due to the complex nonlinear dynamics and the underactuation of this dynamical system. In this article, a new nonlinear optimal control method is proposed for solving the problem of control and stabilization of the 4-DOF planar (XY) inverted pendulum. To apply the proposed nonlinear optimal control method, the dynamic model of the planar inverted pendulum undergoes first approximate linearization around a temporary operating point that is updated at each iteration of the control algorithm. The linearization takes place through first-order Taylor series expansion and through the computation of the Jacobian matrices of the pendulum’s state-space description. For the approximately linearized model of the planar inverted pendulum an H-infinity feedback controller is designed. Actually, the H-infinity controller stands for the solution of the optimal control problem for the planar inverted pendulum under uncertainty and external perturbations. For the computation of the feedback gains of the H-infinity controller an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control algorithm are proven through Lyapunov analysis. The proposed control method achieves fast and accurate tracking of setpoints under moderate variations of the control inputs.