Asymptotic harmonic generator design via modification of Van der Pol oscillator
Periodic Control Systems, Volume # 3 | Part# 1
Y. Orlov; L. T. Aguilar; L. Acho; A. Ortiz
Digital Object Identifier (DOI)
variable structure control,orbital stabilization,Van der Pol equation
A well-known Van der Pol oscillator is modified to be introduced into the synthesis as an asymptotic harmonic generator of the periodic motion. The proposed modification possesses a limit cycle, producing a single harmonic as opposed to multi-harmonics of a standard harmonic oscillator. The parameters of the asymptotic harmonic generator are shown to specify damping, amplitude, and frequency of the limit cycle production. A model orbit stabilization approach to swing up control of a two-link pendulum robot (Pendubot) is used as a case of study. The quasihomogeneous control synthesis is utilized to design a variable structure controller that drives the actuated link of the Pendubot to a periodic reference orbit in finite time. Performance issues of the controller constructed are illustrated in an experimental study.
 The Pendubot user's manual. Mechatronics Systems Inc., Champaign, IL, 1998.  Bartolini G., Ferrara A. and Usai E. (1998). Chattering avoidance by second-order sliding mode control. IEEE Trans. Autom. Contr., 43, 241-246.  Fantoni I., Lozano R. and Spong M. (2000). Energy Based Control of the Pendubot. IEEE Trans. Aut. Contr., 45(4), 725-729.  Filippov A.F. (1988). Differential equations with discontinuous right-hand sides. Dordrecht: Kluwer Academic Publisher.  Fridman L. and Levant A. (2002). Higher order sliding modes," in Sliding mode control in engineering, W. Perruquetti and J.-P. Barbout (eds.), New York: Marcel Dekker, pp. 53-102, 2002.  Hill D. and Moylan P. (1976). The stability of nonlinear dissipative systems. IEEE Trans. Auto. Ctrl., 21, 708- 711.  Khalil H. (2002). Nonlinear systems, third edition, New Jersey: Prentice Hall.  Levant A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control , 58, 1247-1263.  Orlov Y. (2005a). Finite-time stability and robust control synthesis of uncertain switched systems. SIAM Journal on Control and Optimization, 43, pp. 1253-1271.  Orlov Y. (2005b). Finite time stability and quasihomogeneous control synthesis of uncertain switched systems with application to underactuated manipulators. Proc. of the 44th Conference on Decision and Control, Seville, Spain.  Orlov Y., Acho L. and Aguilar L. (2004). Quasihomogeneity approach to the pendubot stabilization around periodic orbits. Proc. 2nd IFAC Symposium on Systems, Structure and Control, Preprints, Oaxaca-Mexico.  Orlov Y., Aguilar L., Acho L. and Ortiz A. (2007). Asymptotic Harmonic Generator and Its Application to Finite Time Orbital Stabilization of a Friction Pendulum with Experimental Verification. To appear in International Journal of Control.  Shiriaev A., Perram J.W. and Canudas-de-Wit C. (2005). Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach. IEEE Trans. Autom. Contr., 50, 1164-1176.  Sira-Ramirez H. (1987). Harmonic response of variable-structure-controlled Van der Pol oscillators. IEEE Trans. Circuits and Systems, 34, 103-106.  Slotine J.-J. and Li W. (1991). Applied Nonlinear Control. New Jersey: Prentice Hall.  Spong M.W. (1995). The Swing Up Contol Problem for the Acrobot. IEEE Control Systems Magazine, 49-55.  Spong M.W. and Praly L. (1997). Control of Underactuated Mechanical Systems Using Switching and Saturation. Lecture Notes in Control and Information Sciences 222, Springer Verlag, London, 163-172.  Utkin V.I., Guldner J. and Shi J. (1999). Sliding modes in Electromechanical Systems. London: Taylor and Francis.  Wang H.-H. and Krstic M. (2000). Extremun Seeking for Limit Cycle Minimization. IEEE Trans. Aut. Contrl., 45(12), 2432-2437.