Pareto Optimal Design of a Fuzzy Adaptive Hierarchical Sliding-mode Controller for an X-Z Inverted Pendulum System

ABSTRACT

The control problem of a three-degree-of-freedom (3-DOF) X-Z inverted pendulum as an unstable, multi-input-multi-output, under-actuated and high-order nonlinear system is facing many challenges. In this paper, a Fuzzy Adaptive Hierarchical Sliding-Mode Controller (FAHSMC) is optimally designed for the X-Z inverted pendulum system utilizing the Multi-Objective Genetic Algorithm (MOGA). To this end, at first, the state variables of this system are converted to the new state variables via five steps of a forward transformation process. In this transformation, the dynamical equations of the X-Z inverted pendulum system are reorganized to an appropriate form for control implementations. Then, the novel FAHSMC is designed for stabilization and tracking control of the system, and the stability analysis of the controller is proved via the Lyapunov stability theory. After that, the state and control variables of the X-Z inverted pendulum are computed through six steps of a backward transformation process. Ultimately, the MOGA is exerted for Pareto optimal design of the proposed control approach applied on the regarded X-Z inverted pendulum system. The tracking error of the cart position in the X-direction, the tracking error of the cart position in the Z-direction and the angle error of the pendulum are selected as three inconsistent objective functions for multi-objective optimization operations. Numerical results and diagrams illustrate that the proposed approaches can accurately converge the system states to the desirable trajectories in a short time and yield superior Pareto optimal fronts in comparison with the Hierarchical Sliding-Mode Controller (HSMC) and the Adaptive Hierarchical Sliding-Mode Controller (AHSMC).

KEYWORDS:

• X-Z inverted pendulum
• Forward transformation
• Backward transformation
• Fuzzy adaptive hierarchical sliding-mode controller
• Multi-objective genetic algorithm
• Pareto optimal design

Additional information

Notes on contributors

R. Abedzadeh Maafi

Rahmat Abedzadeh Maafi received his BSc and MSc degrees in mechanical engineering from Islamic Azad University, Iran in 2008 and 2012, respectively. Now, he is a PhD student in mechanical engineering at Science and Research Branch, Islamic Azad University, Tehran, Iran. His research interests include optimization algorithms, non-linear and robust control, robotics, and computational methods. Email: [email protected]

S. Etemadi Haghighi

Shahram Etemadi Haghighi received his BSc degree in mechanical engineering from Ferdowsi University of Mashad, Mashad, Iran in 2000. He also received his MSc and PhD degrees in mechanical engineering from Sharif University of Technology, Tehran, Iran in 2003 and 2011, respectively. Now, he is an associate professor of Mechanical Engineering at Science and Research Branch, Islamic Azad University, Tehran, Iran. His studies are in the field of non-linear vibration and control and he is interested in swarm robotics and flapping-wing robots. Email: [email protected]

M. J. Mahmoodabadi

Mohammad Javad Mahmoodabadi received his BSc and MSc degrees in mechanical engineering from Shahid Bahonar University of Kerman, Iran in 2005 and 2007, respectively. He received his PhD degree in mechanical engineering from the University of Guilan, Rasht, Iran in 2012. He worked for two years in the Iranian textile industries. During his research, he was a scholar visitor with Robotics and Mechatronics Group, University of Twente, Enschede, the Netherlands for 6 months. Now, he is an associate professor of Mechanical Engineering at the Sirjan University of Technology, Sirjan, Iran. He has published about 150 scientific articles in international journals and conference proceedings. His research interests include optimization algorithms, non-linear and robust control, and computational methods.