Abstract
The design of robust computer control systems for balancing and attitude control of double and triple inverted pendulums is considered in this paper. For the double inverted pendulum, a DC motor mounted at the upper hinge is used to balance and control attitude of the upper link. For the triple inverted pendulum a DC motor mounted at the middle hinge is used to control the middle link, whereas proportional position control applied to a motor at the upper hinge is utilised to maintain the upper link in alignment with the middle link. In both cases the lower hinge is left free to rotate. The controller designs are based on linearised discrete-time models of the inverted pendulums. Each controller utilises state feedback implemented via reduced-order state observers. The relative stability properties of the control systems are evaluated using Nyquist plots of suitably defined functions. The controllers are designed using Matlab and implemented in a PC using C language. Experimental results showed satisfactory performance.
References
| Anderson, B. D. O., Moore, J. B. 1989. Optimal Control: Linear Quadratic Methods, Prentice-Hall, chapters 5 and 8. Google Scholar | |
| Anderson, C. W. 1989. ‘Learning to control an inverted pendulum using neural networks’, IEEE Control Systems Magazine, 9, 31–37. Google Scholar, Crossref | |
| Boyle, D. 1987. ‘Computing rank-deficiency of rectangular matrix pencils’, Systems & Control Letters, 9, 207–214. Google Scholar, Crossref | |
| Cetin, M. 1994 ‘Attitude Control of a Triple Inverted Pendulum’, MSc dissertation, Dept Electronic & Electrical Eng, University of Salford, UK. Google Scholar | |
| Doyle, J. C., Stein, G. 1979. ‘Robustness with observers’, IEEE Trans on Automatic Control, AC-24, 607–611. Google Scholar, Crossref | |
| Eising, R. 1984. ‘Between controllable and uncontrollable’, Systems & Control Letters, 4, 263–264. Google Scholar, Crossref | |
| Furuta, K., Ochiai, T., Ono, N. 1984. ‘Attitude control of a triple inverted pendulum’, Int J Control, 39, 1351–1365. Google Scholar, Crossref | |
| Gahinet, P., Laub, A. J. 1992. ‘Algebraic Riccati Equations and the distance to the nearest uncontrollable pair’, SIAM J Control and Optimization, 30, 765–786. Google Scholar, Crossref | |
| Golliday, C. L., Hemami, H. 1976. ‘Postural stability of the two-degree-of-freedom biped by general linear feedback’, IEEE Trans on Automatic Control, AC-21, 74–79. Google Scholar, Crossref | |
| Hemami, H., Wyman, B. F. 1979. ‘Modeling and control of constrained dynamic systems with application to biped locomotion in the frontal plane’, IEEE Trans on Automatic Control, AC-24, 526–535. Google Scholar, Crossref | |
| Larcombe, P. J. 1992. ‘On the control of a two-dimensional multi-link inverted pendulum: The form of the dynamic equations from choice of co-ordinate system’, Int J Systems Sci, 23, 2265–2289. Google Scholar, Crossref | |
| Maciejowski, J. M. 1989. Multivariable Feedback Design. Addison-Wesley, chapters 3 and 5. Google Scholar | |
| Mansour, M., Schaufelberger, W. 1989. ‘Software and laboratory experiments using computers in control education’, IEEE Control Systems Magazine, 9, 19–24. Google Scholar, Crossref | |
| Mita, T., Yamaguchi, T., Kashiwase, T., Kawase, T. 1984. ‘Realization of a high speed biped using modern control theory’, Int J. Control, 40, 107–119. Google Scholar, Crossref | |
| Mori, S., Nishihara, H., Furuta, K. 1976. ‘Control of an unstable mechanism: Control of pendulum’, Int J Control, 23, 673–692. Google Scholar, Crossref | |
| O'Reilly, J. 1983. Observers for Linear Systems. Academic Press, chapter 8. Google Scholar | |
| Ozguner, U. 1989. ‘Three-course control laboratory sequence’, IEEE Control Systems Magazine, 9, 14–18. Google Scholar, Crossref | |
| Tarokh, M. 1992. ‘Measures for controllability, observability and fixed modes’, IEEE Trans on Automatic control, AC-37, 1268–1273. Google Scholar, Crossref | |
| Wonham, W. M. 1979. Linear Multivariable Control: A Geometric Approach (2nd Edn), Application of Mathematics 10, Springer-Verlag, chapter 0. Google Scholar, Crossref | |
| Yamakawa, T. 1989. ‘Stabilization of an inverted pendulum by a high-speed fuzzy logic controller hardware system’, Fuzzy Sets and Systems, 32, 161–180. Google Scholar, Crossref | |
| Yamakawa, T. 1993. ‘A fuzzy inference engine in non-linear analog mode and its application to a fuzzy logic control’, IEEE Trans on Neural Networks, 4, 496–522. Google Scholar, Crossref, Medline |
