Inverted Pendulum Optimal Control Based on First Principle Model

  • František Dušek
  • Daniel Honc
  • K. Rahul Sharma
  • Libor Havlíček
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 466)


This paper describes the design procedure of nonlinear dynamical model of a real system—inverted pendulum—cart with pendulum. The aim of the paper is to create a mathematical model based on known constructional, mechanical and electrical characteristics of the physical system. Such a model is linearized into standard linear time-invariant state-space model where the input is motor power voltage and the outputs are cart position and pendulum angle. A linear model is used for discrete-time LQ controller design—state variables are estimated and the cart position is controlled with pendulum in upright metastable position.


Inverted pendulum First principle model Optimal control LQ controller 



This research was supported by Institutional support of The Ministry of Education, Youth and Sports of the Czech Republic and SGS project at FEI.


  1. 1.
    Chan, R.P.M., Stol, K.A., Halkyard, C.R.: Review of modelling and control of two-wheeled robots. Ann. Rev. Control 37(1), 89–103 (2013)CrossRefGoogle Scholar
  2. 2.
    Gao, L.J.X., Huang, Q., Du, Q., Duan, X.: Mechanical design and dynamic modeling of a two-wheeled inverted pendulum mobile robot. In: 2007 IEEE International Conference on Automation and Logistics, pp. 1614–1619. IEEE (2007)Google Scholar
  3. 3.
    Ruan, X., Cai, J.: Fuzzy Backstepping controllers for two-wheeled self-balancing robot. In: 2009 International Asia Conference on Informatics in Control, Automation and Robotics, pp. 166–169. IEEE (2009)Google Scholar
  4. 4.
    Haugen, F.: Mathematical Model of a Pendulum on a Cart. TechTeach, Skien (2011)Google Scholar
  5. 5.
    Patenaude, J.: Simple Inverted Pendulum Cart Dynamics Classical Development. Kansas State University, Mechanical and Nuclear Engineering, Manhattan (2007)Google Scholar
  6. 6.
    Technical Guide: Mabuchi Motor. Mabuchi Motor Co., Ltd

Copyright information

© Springer International Publishing Switzerland 2016

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (, which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  • František Dušek
    • 1
  • Daniel Honc
    • 1
  • K. Rahul Sharma
    • 1
  • Libor Havlíček
    • 1
  1. 1.Faculty of Electrical Engineering and Informatics, Department of Process ControlUniversity of PardubicePardubiceCzech Republic

Personalised recommendations