A Near Nonlinear Optimal Controller for The Nonlinear Inverted Pendulum

Abstract—in this paper, a new analytical method to find an optimal high gain controller for the nonlinear inverted pendulum is introduced. This controller is derived based on the closed form solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the cheap control problem. This methodology employs an algebraic equation with parametric coefficients for the systems with scalar internal dynamics and a differential equation for those systems with the internal dynamics of order higher than one. It is shown that for the far initial conditions the proposed controller significantly outperforms the LQR controller while for the near initial conditions; the regulation error is less than that of the closed-loop system with the high gain LQR controller.