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International Journal of Engineering and Manufacturing(IJEM)

ISSN: 2305-3631 (Print), ISSN: 2306-5982 (Online)

IJEM Vol.11, No.2, Apr. 2021

Optimal Controller Design for the System of Ball-on-sphere: The Linear Quadratic Gaussian (LQG) Case

Full Text (PDF, 1016KB), PP.14-30

Author(s)

Usman Mohammed, Tologon Karataev, Omotayo O. Oshiga, Suleiman U. Hussein

Index Terms

Optimality; Controller; Ball-on-sphere; Gaussian; System

Abstract

Control system plays a critical function as one of the essential bedrocks of contemporary social development. Differential equations are time-based equations. The analysis of these equations according to time-domain, is what the theory of modern control is based on. It uses a state-space method which allows direct design in the time-domain. With the state-space method, many controllers can be designed optimally. LQG is one of these controllers. This controller is covered much in the literature. Despite this, not many works cover the ball-on-sphere system. Therefore, the research designed an optimal LQG controller for the system of ball-on-sphere. System dynamics were first investigated and the mathematical model was derived. After that, the system was linearized and then the state-space representation was obtained. Using this representation, the controller was designed and applied to the system for control. The control was done based on the specified desired system performance. Finally, the controller's performance was analyzed. Results obtained showed that the controller met the desired system performance. The controller satisfied the at least 80% performance requirement with θ_x is 82.35% and θ_y is 82.95% less than their respective unregulated settling times. It was also observed that minimizing the total control energy leads to maximizing the total transient energy. Another finding was that all states played role in regulating the controller to the desired system performance. Unfortunately, a settling time (of the ball's angles) of less than 1.00 sec could not be realized. The realized performance is 2.35% and 2.95% more than the desired performance in x and y directions, respectively, for the ball’s angles settling time. This research is significant because it is the first to design an LQG controller for the ball-on-sphere system. Therefore, bridging the existing gap in the literature is the value of this research.

Cite This Paper

Usman Mohammed, Tologon Karataev, Omotayo O. Oshiga, Suleiman U. Hussein, " Optimal Controller Design for the System of Ball-on-sphere: The Linear Quadratic Gaussian (LQG) Case  ", International Journal of Engineering and Manufacturing (IJEM), Vol.11, No.2, pp. 14-30, 2021. DOI: 10.5815/ijem.2021.02.02

Reference

[1]N. S. Nise, CONTROL SYSTEMS ENGINEERING, 6th Edition. California State Polytechnic University, Pomona: John Wiley & Sons, Inc., 2011.

[2]B. Xu, ‘A Comparative Study of PID and LQR Control Strategies Applied to Inverted Pendulum Systems’, Master of Engineering, M.Eng, University of Guelph, Ontario, Canada, 2019.

[3]E. Zakeri, S. A. Moezi, and Y. Bazargan-Lari, ‘Control of a Ball on Sphere System with Adaptive Feedback Linearization method for regulation purpose’, Majlesi J. Mechatron. Eng., vol. 2, no. 3, pp. 23–27, Sep. 2013.

[4]Y.-H. Chang, W.-S. Chan, and C.-W. Chang, ‘T-S Fuzzy Model-Based Adaptive Dynamic Surface Control for Ball and Beam System’, IEEE Trans. Ind. Electron., vol. 60, no. 6, pp. 2251–2263, Jun. 2013, doi: 10.1109/TIE.2012.2192891.

[5]M.-T. Ho, Y.-W. Tu, and H.-S. Lin, ‘Controlling a ball and wheel system using full-state-feedback linearization [Focus on Education]’, IEEE Control Syst., vol. 29, no. 5, pp. 93–101, Oct. 2009, doi: 10.1109/MCS.2009.934085.

[6]A. Buschhaus and S. Schmal, ‘Robolab Reutlingen University’, https://vvl.reutlingen-university.de/homepage/en/index.html. https://vvl.reutlingen-university.de/homepage/en/index.html#demos/ballOnBall/main (accessed Aug. 26, 2020).

[7]A. D. Usman, A. M. Yusuf, A. Umar, and A. Daniel, ‘Structual analysis of ball-on-sphere system using bond graph technique’, in 2017 IEEE 3rd International Conference on Electro-Technology for National Development (NIGERCON), Owerri, Nov. 2017, pp. 519–524, doi: 10.1109/NIGERCON.2017.8281921.

[8]M.-T. Ho, Y. Rizal, and W.-S. Cheng, ‘Stabilization of a Vision-Based Ball-On-Sphere System’, in 2013 IEEE International Conference on Control Applications (CCA), Part of 2013 IEEE Multi-Conference on Systems and Control, Hyderabad, India, Aug. 2013, pp. 929–934, doi: 10.1109/CCA.2013.6662870.

[9]A. Umar, Z. Haruna, U. Musa, S. A. Mohammed, and M. O. Muyideen, ‘Graphical User Interface (GUI) for Position and Trajectory Tracking Control of the Ball and Plate System Using H-Infinity Controller’, doi: 10.20370/YHAS-N460.

[10]H. Jafari, A. Rahimpour, and M. Pourrahim, ‘Linear Quadratic Gaussian Control for ball and plate system’, in 2012 international conference on computer, control, education and management, Dubai, United Arab Emirates, Jul. 2012, pp. 1–7.

[11]W. Favoreell and B. D. Moor, ‘Model-free subspace-based LQG-design’, in Proceedings of the American Control Conference (ACC), San Diego, California, Jun. 1999, pp. 3372–3376, doi: 10.1109/ACC.1999.782390.

[12]R. Banerjee and A. Pal, ‘Stabilization of Inverted Pendulum on Cart Based on LQG Optimal Control’, in 2018 International Conference on Circuits and Systems in Digital Enterprise Technology (ICCSDET), Kottayam, India, Dec. 2018, pp. 1–4, doi: 10.1109/ICCSDET.2018.8821243.

[13]A. Besancon-Voda, G. Filardi, D. Rey, and A. Franco, ‘LQG optimal control strategies for an electro pneumatic actuator’, in 2001 European Control Conference (ECC), Porto, Sep. 2001, pp. 2670–2675, doi: 10.23919/ECC.2001.7076333.

[14]Q. Jin, S. Ren, and Ling Quan, ‘LQG optimum controller design and simulation base on inter model control theory’, in 2009 IEEE International Conference on Intelligent Computing and Intelligent Systems, Shanghai, China, Nov. 2009, pp. 62–65, doi: 10.1109/ICICISYS.2009.5358234.

[15]R. Soni and Sathans, ‘Optimal control of a ball and beam system through LQR and LQG’, in 2018 2nd International Conference on Inventive Systems and Control (ICISC), Coimbatore, Jan. 2018, pp. 179–184, doi: 10.1109/ICISC.2018.8399060.

[16]J. Vlk and P. Chudy, ‘General aviation digital autopilot design based on LQR/LQG control strategy’, in 2017 IEEE/AIAA 36th Digital Avionics Systems Conference (DASC), St. Petersburg, FL, Sep. 2017, pp. 1–9, doi: 10.1109/DASC.2017.8102058.

[17]M. Ho, Y. Tu, and H. Lin, ‘Controlling a ball and wheel system using full-state-feedback linearization [Focus on Education]’, Control Syst. IEEE, vol. 29, pp. 93–101, Nov. 2009, doi: 10.1109/MCS.2009.934085.

[18]M. Moarref, M. Saadat, and G. Vossoughi, ‘Mechatronic design and position control of a novel ball and plate system’, in 2008 16th Mediterranean Conference on Control and Automation, Ajaccio, France, Jun. 2008, pp. 1071–1076, doi: 10.1109/MED.2008.4602212.

[19]S. A. Moezi, E. Zakeri, Y. Bazargan-Lari, and M. Khalghollah, ‘Fuzzy Logic Control of a Ball on Sphere System’, Adv. Fuzzy Syst., vol. 2014, pp. 1–6, 2014, doi: 10.1155/2014/291430.

[20]S.-Y. Liu, Y. Rizal, and M.-T. Ho, ‘Stabilization of a Ball and Sphere System Using Feedback Linearization and Sliding Mode Control’, in 2011 8th Asian Control Conference (ASCC), May 2011, p. 6.

[21]E. Zakeri, A. Ghahramani, and S. Moezi, ‘Adaptive Feedback Linearization Control of a Ball on Sphere System’, in International Conference on Mechanical Engineering and Advanced Technology, ICMEAT 2012, Isfahan, Iran, Oct. 2012, pp. 1–5.

[22]A. Tewari, Modern control design with MATLAB and SIMULINK. New York: John Wiley, 2002.

[23]G. Welch and G. Bishop, ‘An Introduction to the Kalman Filter’. 2006.

[24]R. Eide, P. M. Egelid, A. Stamsø, and H. R. Karimi, ‘LQG Control Design for Balancing an Inverted Pendulum Mobile Robot’, Intell. Control Autom., vol. 02, no. 02, pp. 160–166, 2011, doi: 10.4236/ica.2011.22019.

[25]R. S. Burns, Advanced control engineering. Oxford, Boston: Butterworth-Heinemann, 2001.

[26]U. Mohammed, S. U. Hussein, M. Usman, and S. Thomas, ‘Design of an Optimal Linear Quadratic Regulator (LQR) Controller for the Ball-On-Sphere System’, International Journal of Engineering and Manufacturing(IJEM), vol. 10, no. 3, pp. 56–70, Jun. 2020, doi: 10.5815/ijem.2020.03.05.