Control Theory Application for Swing Up and Stabilisation of Rotating Inverted Pendulum

Abstract

This paper introduces a new scheme for sliding mode control using symmetry principles for a rotating inverted pendulum, with the possibility of extension of this control scheme to other dynamic systems. This was proven for swing up and stabilisation control problems via the new sliding mode control scheme using both simulations and experiments of rotary inverted pendulum (RIP) underactuated systems. According to the Lyapunov theory, a section of the pendulum was compensated with a scale error in the upright position, as the desired trajectory was followed by the pendulum arm section. As the RIP’s dynamic equations were nonlinearly complex and coupled, the complex internal dynamics made the task of controller design difficult. The system control for the pathway of the reference model of the rotational actuator with the application of the sliding mode technique for moving back and forth up the inverted pendulum’s structure, till the arm to reach the linear range round the vertical upright position, was created and tested in an existent device. The stabilisation scheme was switched on in the sliding mode as soon as the arm reached the linear range. A comparison of the stabilisation performance for the same rotating inverted pendulum as discussed by other authors revealed that the proposed controller was more flexible and reliable in terms of the swing up and stabilisation time.

The Regulation of an Electric Oven and an Inverted Pendulum

“…is the force applied to the mathematical model, which is the regulation variable. In this case, the generalized variables are 𝑥 and 𝜃; therefore, 𝑞 = (𝑥, 𝜃) , and we obtain the elements of each Lagrange equation as follows: The mathematical model of the inverted pendulum is derived from the Euler-Lagrange equations [19][20][21][22][23]. More details of the mathematical model can be found in [19,20,23,24].…”

“…One of the ways to reduce this chattering is to increase the order of the regulator, where the integral of 𝜎 is used. The second-order sliding-mode regulator is defined in Equation (19) [10][11][12].…”

“…The transfer function of the second-order sliding-mode regulator (19) is represented in Equation (21).…”

“…where the high-frequency sign function is within the integral; the regulator gains c and b for the mathematical model of the electric oven are derived from Equation ( 4). The gains are selected by the transfer function of the second-order sliding-mode regulator (19), and the sign-mapping sign(g(t)) is changed by one-unit step-mapping h(t) as follows sign(g(t)) = 2h(t) − 1.…”

“…The linearization method is known as an approximate linearization, which is based on a Taylor series expansion to represent an asymmetrical non-linear mathematical model as a linear mathematical model. This method is also known as Jacobian linearization because of the Jacobian matrices that are used for the linear approximation [19,20,23].…”