Stabilization of Inverted Cart-Pendulum System Using Controller: A Frequency-Domain Approach

This paper focuses on the angular stabilization of inverted cart-pendulum system using controller. The tuning of controller is formulated as a nonlinear optimization problem, in which the objective function is composed of five design conditions in frequency domain. Particle swarm optimization technique has been used for optimizing parameters. Also a PID controller has been designed based on same specifications, and a comparative study has been carried out. All the responses have been calculated using FOMCON toolbox of Matlab/Simulink. 1. Introduction In recent years, Fractional-order PID ( ) controller has become a good generalization of standard PID controller using fractional calculus. Compared to PID controller, the tuning of is more complex and remains a challenging problem. As the PID controller is one of the most commonly used controllers in industries, and in recent years fractional calculus [1–5] has been used to present the more generalized form of PID controller, that is, controller. It is well known that inverted cart-pendulum [6] is an example of underactuated, nonminimum phase and highly unstable system. Because of this, it is very difficult to design a controller for a system like inverted pendulum. The design becomes more difficult because of the physical constraints on track length, applied voltage, and the pendulum angle. Podlubny has proposed a generalization of the PID controller as controller. He also demonstrated that the fractional order PID controller has better response than classical PID controller [1, 7]. Also, many valuable studies have been done for fractional order controllers and their implementations [8–14]. Tuning of the controller using the frequency-domain approaches is studied in many papers. For example, [15] propose a method based on optimization strategies. Tuning of H∞ controllers for fractional single-input single-output (SISO) system was suggested in [12]. In [16], tuning method for based on Ziegler-Nichols and Astrom-Hagglund tuning rules is presented. In this study, a controller has been designed for an unstable system (inverted cart-pendulum system) using frequency-domain approach, and particle swarm optimization (PSO) has been used for obtaining the parameters ( , , , , ). This work is organized as follows: Section 2 covers inverted cart-pendulum system, Section 3 covers FOPID controller, Section 4 presents particle swarm optimization, Section 5 presents FOPID controller design, Section 6 presents simulation results, and the conclusion is provided in Section 7. 2. Inverted Cart-Pendulum System The inverted