Citation: Maraslidis, G.S.; Kottas, T.L.;Tsipouras, M.G.; Fragulis, G.F. Design of a Fuzzy Logic Controller for the Double Pendulum Inverted on a Cart. Information 2022, 13, 379. https://doi.org/10.3390/ info13080379 Academic Editor: Arkaitz Zubiaga Received: 3 July 2022 Accepted: 2 August 2022 Published: 8 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). information Article Design of a Fuzzy Logic Controller for the Double Pendulum Inverted on a Cart George S. Maraslidis , Theodore L. Kottas , Markos G. Tsipouras and George F. Fragulis * Department of Electrical and Computer Engineering, University of Western Macedonia, 50100 Kozani, Greece * Correspondence: gfragulis@uowm.gr Abstract: The double-inverted pendulum (DIP) constitutes a classical problem in mechanics, whereas the control methods for stabilizing around the equilibrium positions represent the classic standards of control system theory and various control methods in robotics. For instance, it functions as a typical model for the calculation and stability of walking robots. The present study depicts the controlling of a double-inverted pendulum (DIP) on a cart using a fuzzy logic controller (FLC). A linear-quadratic controller (LQR) was used as a benchmark to assess the effectiveness of our method, and the results showed that the proposed FLC can perform significantly better than the LQR under a variety of initial system conditions. This performance is considered very important when the reduction of the peak system output is concerned. The proposed controller equilibration and velocity tracking performance were explored through simulation, and the results obtained point to the validity of the control method. Keywords: double-inverted pendulum; linear quadratic regulators; fuzzy logic controllers; automatic control 1. Introduction Modern control theory flourished during the 1960s, principally due to the development of aerospace dynamics and the emergence of space exploration. Today, automated control systems (ACS) [1] are implemented telecommunications, medicine, economics, power and electrical systems, and biology. Emerging trends of mechatronics, specifically robotics, ref. [2] exploit these mathematical possibilities. Although robotics and informatics have revolutionized commercial enterprises, the need for better and effective control challenged scientists to develop new mathematical models and concepts such as fuzzy logic, machine learning and neural networks to constrain the system closer to human thinking. Fuzzy logic is a new theory of mathematics, an extension of classical binary logic [3] proposed by the mathematician Lotfi Zadeh in his study of fuzzy sets. Zadeh’s theory was based on a study conducted by Lukasiewicz and Tarski back in the 1920s. The first fuzzy logic controller was developed, manufactured, and integrated into electronic and electrical equipment, such as cameras and washing machines, in Japan in the 1980s [3–6]. The results were rather encouraging and led to further acceptance of the theory. Intelligent navigation systems can now replace human pilots in helicopters with amazing performance [7,8]. Integrating these controls into difficult and complex systems that previously only humans could operate is very advantageous. Moreover, engineers cooperate with experts to design fuzzy rule-based systems and develop efficient models. In this way, complex processes can be analyzed, and elements closer to human behavior can be found. The descriptor system found in nature is an example of a fuzzy logic controller test case. In this study a double- inverted pendulum on a cart, a classical engineering problem [9–11], was used to evaluate a fuzzy logic controller (FLC) compared to a method using a linear-quadratic control (LQR), which will be discussed and analyzed [12–14]. The purpose of this paper is to strengthen the arguments in favor of an FLC and to demonstrate its potential and the improvements it can offer to existing control methods for complex physical systems in nature that show Information 2022, 13, 379. https://doi.org/10.3390/info13080379 https://www.mdpi.com/journal/information