Backstepping controller design with a quadratic error for a double inverted pendulum

This study aims at presenting a conceptualisation of a double inverted pendulum system based on modelling and controlling this model by applying only a single torque in the upper part instead of the lower part of the double inverted pendulum. Nonlinear dynamic equations were analysed using the Lagrangian dynamic formulation, and the graphical presentation of the system has been achieved through applying the bond graphs approach. The double inverted pendulum representation was incarnated using the 20-Sim software. To achieve the system's stability, only one command was applied on the upper part. Most important, a control approach combining the backstepping method with the quadratic error was designed. Indeed, the simulation results affirmed the effectiveness of the backstepping controller with the quadratic error, and good response of the system's flexibility was ensured in the sense that it can be adjusted from the initial position to the equilibrium position.