LMI Based Robust Control of Inverted Pendulum System

  • Md. Shah AlamEmail author
  • Sharmistha Mandal
Conference paper
Part of the Learning and Analytics in Intelligent Systems book series (LAIS, volume 12)


In present study, a state feedback robust controller has been designed for an inverted pendulum along with its driving servo-system. The objective is to stabilize extremely unstable and non-linear servo-driven inverted pendulum system. The controller has been designed using Linear Matrix Inequality (LMI) technique by placing the closed-loop poles in a selected region. The controller gives satisfactory result in both stabilization and tracking problems with minimum settling time, maximum overshoot and undershoots. For robustness, the controller is tested in presence of external noise and parameter variations. The performance of designed state-feedback controller is compared with the performance of Linear Quadratic Regulator (LQR) controller. The proposed controller gives superior result in terms of transient response specifications and noise elimination.


Inverted pendulum system Regional pole-placement Linear Matrix Inequality Robust control 


  1. 1.
    Ogata, K.: Modern Control Engineering, 4th edn. Pearson Education, Upper Saddle River (2002)zbMATHGoogle Scholar
  2. 2.
    Muskinja, N., Tovomik, B.: Swinging up and stabilization of a real inverted pendulum. IEEE Trans. Industr. Electron. 53(2), 631–639 (2006)CrossRefGoogle Scholar
  3. 3.
    Kennedy, E., Tran, H.: Swing-up of an inverted pendulum on a cart using a modified energy based approach. In: Proceedings of the International Multi Conference of Engineers and Computer Scientists, IMECS 2016, Hong Kong, 16–18 March, vol. I (2016)Google Scholar
  4. 4.
    Valluru, S., Singh, M.: Stabilization of nonlinear inverted pendulum system using MOGA and APSO tuned nonlinear PID controller. Cogent Eng. 4(1), 1–15 (2017)CrossRefGoogle Scholar
  5. 5.
    Jiang, S., Li, M., Wang, C.: Design and simulation of fractional order PID controller for an inverted pendulum system. In: IEEE International Conference on Manipulation, Manufacturing, and Measurement on the Nano Scale (3M-NANO), Shanghai, 7–11 August 2017, pp. 349–352 (2017)Google Scholar
  6. 6.
    Bakarac, P., Klauco, M., Fikar, M.: Comparison of inverted pendulum stabilization with PID, LQ, and MPC control. In: Proceedings of the 29th International Conference 2018 Cybernetics & Informatics (K&I), Lazy pod Makytou, Slovakia, 31 January–3 February 2018 (2018)Google Scholar
  7. 7.
    Apkarian, J., Lacheray, H., Martin, P.: Instructor Workbook of Linear Inverted Pendulum Experiment. Standardized for ABET Evaluation Criteria. Quanser Inc., Ontario (2012)Google Scholar
  8. 8.
    Gahinet, P., Nemirovski, A., Laub, A., Chilali, M.: The LMI Control Toolbox for Use with MATLAB. The Mathworks, Inc., Natick (1995)Google Scholar
  9. 9.
    Chilali, M., Gahinet, P.: H design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41(3), 358–367 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Scherer, C., Gahinet, P., Chilali, M.: Multi-objective output-feedback control via LMI optimization. IEEE Trans. Autom. Control 42(7), 896–911 (1997)CrossRefGoogle Scholar
  11. 11.
    Mandal, S., Sutradhar, A.: LMI based robust blood glucose regulation in Type-1 diabetes patient with multi-meal ingestion. J. Inst. Eng. (India) Ser. B 95(2), 121–128 (2014)CrossRefGoogle Scholar
  12. 12.
    Mandal, S., Sutradhar, A.: Multi-objective control of blood glucose with H and pole-placement constraints. Int. J. Dyn. Control 5(2), 357–366 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mandal, S., Sutradhar, A.: Robust multi-objective blood glucose control in Type-1 diabetic patient. IET Syst. Biol. 13(3), 135–146 (2019)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Aliah UniversityDepartment of Electrical EngineeringKolkataIndia

Personalised recommendations