DYNAMIC MATHEMATICAL MODELING AND CONTROL ALGORITHMS DESIGN OF AN INVERTED PENDULUM SYSTEM

Year 2019, Volume: 3 Issue: 1, 14 - 24, 01.01.2019

Ayodeji Okubanjo , Oluwadamilola Oyetola

https://doi.org/10.31127/tuje.435028

Cited By: 3

Abstract

The inimitable features of multivariable, instability, non-minimum phase and non-linearity has established an inverted pendulum system as benchmark to investigate and test new emerging control schemes. In this paper, the objectives are to explicitly model the system dynamics of an inverted pendulum and implement different control algorithms that will stabilize the pendulum in the upright vertical position by controlling the input force applied to the cart in the horizontal position. The mathematical model is derived based on the energy property of Lagrange approach and the control algorithms are expanded on the derived mathematical model in MATLAB-SIMULINK environment. Hence, we proposed four different controls algorithms proportional-integral-derivative controller (PID), pole placement feedback controller (PPFC), linear quadratic regulator controller (LQR) and linear quadratic regulator with estimator (LQR+Estimator) for the control of the linearized inverted pendulum system. The study compares the proposed control algorithms in terms of system response and performance.

Keywords

Mathematical modeling, Control algorithms, Inverted pendulum, Euler-Lagrange, Simulink

References

• Ababneh, M., Salah, M., & Alwidyan, K. (2011). Linearization of Nonlinear Dynamical Systems : A Comparative Study. Jordan Journal of Mechanical and Industrial Engineering, 5(6), 567–571.
• Agarana, M. C., & Ajayi, O. O. (2017). Dynamic Modeling and Analysis of Inverted Pendulum using Lagrangian-Differential Transform Method. In Proceedings of the World Congress on Engineering 2017 Vol II WCE 2017 (Vol. II, pp. 3–8).
• Bettayeb, M., Boussalem, C., Mansouri, R., & Al-saggaf, U. M. (2014). Stabilization of an inverted pendulum-cart system by fractional PI-state feedback. ISA Transactions, 53(2), 508–516.
• Bogdanov, A. (2004). Optimal control of a double inverted pendulum on a cart. CSEE, OGI School of Science and Engineering, …. Retrieved from http://speech.bme.ogi.edu/publications/ps/bogdanov04a.pdf%5Cnfile:///C:/Users/Spencer/Documents/Mendeley Desktop/Optimal control of a double inverted pendulum on a cart - 2004 - Bogdanov.pdf
• Chandan, Kumar, Santosh, Lal, Nilanjan, Patra, Kaushik, Halder, Motahar, R. et . a. (2012). Optimal Controller Design for Inverted Pendulum System based on LQR method 1,2,3,4. In 2012 IEEE International Conference on Advanced Communication Control and Computing Technologies (ICACCCT) (pp. 259–263).
• Chen, X., Yu, R., Huang, K., & Zhen, S. (2018). Simulation Modelling Practice and Theory Linear motor driven double inverted pendulum : A novel mechanical design as a testb e d for control algorithms. Simulation Modelling Practice and Theory, 81, 31–50.
• Dastranj, M. R., Moghaddas, M., Ghezi, Y., & Rouhani, M. (2012). Robust Control of Inverted Pendulum Using Fuzzy Sliding Mode Control and Genetic Algorithm. International Journal of Information and Electronics Engineering, 2(5), 773–776.
• Eizadiyan, M. A., & Naseriyan, M. (2015). Control Of Inverted Pendulum Cart System by Use of M d F m x F m y d x x, 27(2), 1063–1068.
• Goswami, A. (2013). The analysis of inverted pendulum control and its other applications. Journal of Applied Mathematics & Bioinformatics, 3(3), 113–122.
• Guo, H., & Unversity, S. L. (n.d.). Modelling and Simulation of a Single Inverted Pendulum System Based on Matlab, 463–464.
• İlgen, S., Oflaz, E., Gülbahçe, E., & Çakan, A. (2016). Applied Mathematics , Electronics and Computers Modelling and Control of a Single-Wheel Inverted Pendulum by Using Adams and Matlab. International Journal of Applied Mathematics, Electronics and Computers, 4, 326–328.
• Ilyas, A., Yahya, S., & Al-rizzo, H. (2017). Fuzzy-logic control of an inverted pendulum on a cart R. Computers and Electrical Engineering, 61, 31–47.
• Irfan, Jamil, Rehan, Jamil, Zhao, Jinquan, Rizwan, Jamil& Abdus, S. (2013). Mathematical Model Analysis And Control Algorithms Design Based On State Feedback Method of Rotary Inverted Pendulum. International Journal of Research In Engineering & Technology, 1(3), 41–50.
• Jose, A., Augustine, C., Malola, S. M., & Chacko, K. (2015). Performance Study of PID Controller and LQR Technique for Inverted Pendulum. World Journal of Engineering and Technology, (May), 76–81.
• Kafetzis, I., Moysis, L., & Sciences, M. (2017). Inverted Pendulum : A system with innumerable applications 2 . Modeling the Inverted Pendulum Consider the system consisting of a cart with a rod placed on its center as. In IKEECONF-2017 9th International Week Dedicated to MathsAt: Thessaloniki, Greece.
• Krishna, N. R., Bindhu, K. R., & Vinod, B. R. (2016). Modeling and controller design of cart inverted pendulum system using MRAC scheme. Frontiers of Current Trends In Engineering And Technology April, 24(April), 21–24.
• Kumar, R., Singh, R. B., & Das, J. (2013). Modeling And Simulation of Inverted Pendulum System Using Matlab : Overview. International Journal of Mechanical and Production Engineering, 1(4), 52–55.
• Lee, G. H., Lee, H. J., Choi, H. J., Jeon, H. J., & Jung, S. (2009). Application of Mobile Inverted Pendulum Systems to Boxingbots for a Boxing Game. In 2009 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Suntec Convention and Exhibition Center Singapore (pp. 963–968).
• Lee, J., Mukherjee, R., & Khalil, H. K. (2015). Automatica Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties ✩. Automatica, 54, 146–157.
• Mishra, S. K., & Chandra, D. (2014). Stabilization and Tracking Control of Inverted Pendulum Using Fractional Order PID Controllers. Journal of Engineering.
• Nithya, R., & Vivekanandan, C. (2014). Stability Analysis and State Feedback Stabilization of Inverted Pendulum. International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN, 3(5), 310–320. Retrieved from www.ijcsmc.com
• Norman S, N. (2011). Control Systems Engineering (sixth). River Street Hoboken: Sons, John Wiley & Sons.
• Oltean, S.-E. (2014). Swing-up and Stabilization of the Rotational Inverted Pendulum Using PD and Fuzzy-PD Controllers. Procedia Technology, 12, 57–64.
• Prado, S. D., & Fernandes, H. A. (2017). Commun Nonlinear Sci Numer Simulat A new look on the stabilization of inverted pendulum with parametric excitation and large random frequencies : Analytical and numerical approaches. Communications in Nonlinear Science and Numerical Simulation, 51, 105–114.
• Prayitno, A., Indrawati, V., & Trusulaw, I. I. (2017). Optimal control of inverted pendulum system using PID controller , LQR and MPC Optimal control of inverted pendulum system using PID controller , LQR and MPC. In 14th ICSET-2017 IOP Conf. Series: Materials Science and Engineering.
• Přemysl, Strakoš, Jiří, T. (2017). Mathematical Modelling and Controller Design of Inverted Pendulum. In Carpathianian Control Conference (ICCC) 18th International, pp. 388–393.
• Sangfeel, K., Eunji, S., Kyungsik, K., & Byungseop, S. (2015). Design of Fuzzy Logic Controller for Inverted Pendulum-type Mobile Robot using Smart In-Wheel Motor. Indian Journal of Science and Technology, 8 (April), 493–503.
• Singh, A. K., & Ph, D. (2015). Design of a Robust Controller for Inverted Pendulum. International Journal of Computer Applications, 112(16), 23–28.
• Siradjuddin, I., Setiawan, B., Fahmi, A., Amalia, Z., & Rohadi, E. (2017). linearised model. International Journal of Mechanical & Mechatronics Engineering IJMMEIJENS, 17(1), 119–126.
• Urrea, C., & Pascal, J. (2017). Parameter identification methods for real redundant manipulators. Journal of Applied Research and Technology, 15, 320–331.
• Wang, J. (2011). Simulation Modelling Practice and Theory Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory, 19(1), 440–449.