I. Introduction
Many studies have investigated the dynamics of inverted pendulums [1]–[10]. The primary objective of this study is to achieve pendulum balance when a pendulum is upright. Such a system can be applied as a primary tool for examining the balance achieved with active balancing systems, and is an important contribution to the development legged machines and robotic locomotion studies. An inverted pendulum is a typical unstable complex nonlinear system widely utilized for educational purposes and control research. To date, most control experiments use the rail-cart structure. This inverted pendulum consists of a cart with an attached pendulum running along a straight rail. The control objective is to move the pendulum to the upper unstable equilibrium position by moving the cart along the rail. Clearly, such an inverted pendulum is a one-dimensional control structure.