Linear Quadratic Optimal Control of an Inverted Pendulum on a Cart using Artificial Bee Colony Algorithm: An Experimental Study

Year 2017, Volume: 32 Issue: 2, 109 - 124, 15.06.2017

Barış Ata Ramazan Çoban

https://doi.org/10.21605/cukurovaummfd.358391

Cited By: 1

Abstract

This study presents a Linear Quadratic Optimal (LQR) controller design for an inverted pendulum on a
cart using the Artificial Bee Colony (ABC) algorithm. Main design parameters of the linear quadratic
regulator are the weighting matrices. Generally, selecting weighting matrices is managed by trial and
error since there exists no apparent connection between these weights and time domain requirements such
as settling time, steady state error, and overshoot percentage. In this study after deriving the mathematical
models of the inverted pendulum on a cart and the DC motor, an LQR controller is designed using the
ABC algorithm to determine weighting matrices to overcome LQR design difficulties. The comparison
and experimental results justify that the ABC algorithm is a very efficient way to determine LQR
weighting matrices in comparison with a method in literature. 

Keywords

Artificial bee colony algorithm, Linear quadratic regulator, Inverted pendulum

Yapay Arı Kolonisi Algoritması ile Bir Arabalı Ters Sarkacın Lineer Kuadratik Kontrolü: Deneysel Bir Çalışma

Abstract

Bu çalışmada, Lineer Kuadratik Regülatör (LQR) ile bir ters sarkacın kontrolü için, Yapay Arı Kolonisi
(ABC) optimizasyon algoritmasına dayalı bir metot önerilmiştir. LQR’ın temel tasarım parametreleri
ağırlık matrisleridir. Ağırlık matrislerinin değerleri ile yüzde aşımı, yerleşme zamanı ve kararlı hal hatası
gibi zaman uzayı performans kriterleri arasında doğrudan bir ilişki olmadığı için bu matrislerin seçimi
genellikle deneme yanılma yöntemiyle gerçekleştirilmektedir. Bu çalışmada arabalı ters sarkaç ve bu
mekanizmayı hareket ettiren DC motorun matematiksel modellerinin elde edilmesinin ardından sürü
zekası temelli bir optimizasyon algoritması olan ABC algoritması kullanılarak bir LQR kontrolör
tasarlanmıştır. Karşılaştırma ve deney sonuçları, ABC algoritmasının literatürde önerilen bir yöntem ile
karşılaştırıldığında ağırlık matrislerinin belirlenmesinde daha etkin bir yol olduğunu göstermiştir.

Keywords

Yapay arı kolonisi algoritması, Lineer kuadratik regülatör, Ters sarkaç

References

• 1. Anderson, C. W., 1989. Learning to Control an Inverted Pendulum Using Neural Networks, IEEE Control Systems Magazine, 9(3), 31-37.
• 2. Kuo, A. D, 2007. The Six Determinants of Gait and the Inverted Pendulum Analogy: A Dynamic Walking Perspective, Human Movement Science, 26(4), 617-656.
• 3. Jeong, S., Takahashi, T., 2008. Wheeled Inverted Pendulum Type Assistant Robot: Design Concept and Mobile Control, Intelligent Service Robotics, 1(4), 313-320.
• 4. Feedback Instruments Ltd., 2006. 33-936s Digital Pendulum Control Experiments Manual.
• 5. Kalman, R. E., 1964. When is a Linear Control System Optimal?, Journal of Basic Engineering, 86(1), 51-60.
• 6. Kwakernaak, H., Sivan, R., 1972. Linear Optimal Control Systems, New York.
• 7. Fonseca Neto, J.V., Abreu, I. S., Da Silva, F. N., 2010. Neural–genetic Synthesis for StateSpace Controllers Based on Linear Quadratic Regulator Design for Eigenstructure Assignment, IEEE Transactions on Systems, Man, and Cybernetics, 40(2), 266-285.
• 8. Yaoqing, W., 1992. The Determination of Weighting Matrices in lq Optimal Control Systems, Acta Automatica Sinica, 2(11).
• 9. Bryson, A. E., 1975. Applied Optimal Control: Optimization, Estimation and Control, New York.
• 10. Ghosh, A., Krishnan, T., Subudhi, B., 2012. Brief Paper-robust Proportional-integralDerivative Compensation of an Inverted CartPendulum System: An Experimental Study, IET Control Theory & Applications, 6(8), 1145-1152.
• 11.Bottura, C. P., Fonseca Neto, J. V., 2000. RuleBased Decision-making Unit for Eigenstructure Assignment Via Parallel Genetic Algorithm and Lqr Designs, American Control Conforence Proceedings, 467-471.
• 12. Mobayen, S., Rabiei, A., Moradi, M., Mohammady, B., 2011. Linear Quadratic Optimal Control System Design using Particle Swarm Optimization Algorithm, International Journal of Physical Sciences, 6(30), 6958- 6966.
• 13. Ata, B., Coban, R., 2015. Artificial Bee Colony Algorithm Based Linear Quadratic Optimal Controller Design for a Nonlinear Inverted Pendulum, International Journal of International Systems. and Applications in Engineering, 3(1), 1-6.
• 14. Karaboga, D., 2005. An Idea Based on Honey Bee Swarm for Numerical Optimization, Technical Report TR06, Erciyes University, Kayseri.
• 15. Karaboga, D., Basturk, B., 2007. A Powerful and Efficient Algorithm for Numerical Function Optimization: Artificial Bee Colony Algorithm, Journal of Global Optimization, 39(3), 459–471.
• 16. Ercin, O., Coban, R., 2012. Identification of Linear Dynamic Systems using Artificial Bee Colony Algorithm, Turkish Journal of Electrical Eng. and Computer Sciences, 20(1), 1175–1188.
• 17. Nise, N. S., 2011. Control Systems Engineering, USA.
• 18. Mablekos, V. E., 1980. Electric Machine Theory for Power Engineers, New York.
• 19. Ogata, K., 2001. Modern Control Engineering, New Jersey.
• 20. Karaboga D., Akay, B., 2009. A Comparative Study of Artificial Bee Colony Algorithm, Applied Mathematics and Computation, 214(1), 108–132.
• 21.Coban, R., 2011. A fuzzy Controller Design for Nuclear Research Reactors using the Particle Swarm Optimization Algorithm, Nuclear Eng. and Design, 241(5), 1899–1908.
• 22. Messner, W., Tilbury, D., Inverted Pendulum: State-space Methods for Controller Design, http://ctms.engin.umich.edu/CTMS/index.php? example=InvertedPendulum§ion=ControlS tateSpace , Accessed:02:12:2016.
• 23. Franklin, G. F., 1997. Powell, D. J., Workman, M. L., Digital Control of Dynamic Systems. UK.
• 24. Ayres, F., 1976. Matrices, New York.