Abstract
Three methods are investigated for the tracking problem of the famous cart–pole system (a kind of planar inverted pendulum). The output is required to track a sinusoid signal. Control design is based on the linearized model. First, we show that using output error and states feedback, approximate tracking can be achieved with bounded tracking error. Then exact tracking via output regulation is investigated. By constructing a regulator equation, the equivalent input and equivalent states which are needed to maintain output at the reference trajectory can be calculated. We show that the tracking problem is equivalent to the stabilizing problem in the states error coordinate. Finally, we study exact tracking via stable system center method. Because of the nonminimum phase property, a bounded solution for the internal dynamics is required and is estimated by stable system center method. Then the tracking problem can also be transformed into a stabilizing problem. Simulations are made for each method.
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References
Hunt LR, Meyer G (1997) Stable inversion for nonlinear systems. Automatica 33(8):1549–1554
Devasia S, Chen D, Paden B (1996) Nonlinear inversion-based output tracking. Autom Control IEEE Trans 41(7):930–942
Byrnes CI, Priscoli FD, Isidori A (1997) Output regulation of nonlinear systems. In: Output regulation of uncertain nonlinear systems. Birkhäuser Boston, pp 131−140
Jie H, Chen Z (2005) A general framework for tackling the output regulation problem. IEEE Trans Autom Control 49(12):2203–2218
Isidori A (1999) Nonlinear control systems II. Springer, London
Shkolnikov IA, Shtessel YB (2001) Tracking controller design for a class of nonminimum-phase systems via the method of system center. IEEE Trans Autom Control 46(10):1639–1643
Shkolnikov I, Shtessel A et al (2002) Tracking in a class of nonminimum-phase systems with nonlinear internal dynamics via sliding mode control using method of system center. Automatica 38(5):837–842
Landry M, Campbell SA, Morris K et al (2005) Dynamics of an inverted pendulum with delayed feedback control. SIAM J Appl Dyn Syst 4(2):333–351
Angeli D (1999) Almost global stabilization of the inverted pendulum via continuous state feedback. Automatica 37(01):1103–1108
Shiriaev A, Pogromsky A, Ludvigsen H et al (2000) On global properties of passivity-based control of an inverted pendulum. Int J Robust Nonlinear Control 3(4):2513–2518 vol.3
Lozano R, Fantoni I, Dan JB (2000) Stabilization of the inverted pendulum around its homoclinic orbit. Syst Control Lett 40(3):197–204
Olfati-Saber R (2010) Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum. In: Proceedings of the, IEEE conference on decision and control, pp 1174–1181
Bedrossian NS (1992) Approximate feedback linearization: the cart-pole example. In: Proceedings of IEEE International Conference on Robotics and Automation 1987-1992, vol 3. IEEE
Gurumoorthy R, Sanders SR (1993) Controlling non-minimum phase nonlinear systems—the inverted pendulum on a cart example. In: American control conference, pp 680−685
Yan Q (2004) Output tracking of underactuated rotary inverted pendulum by nonlinear controller. In: IEEE conference on decision & control, vol. 3, pp 2395−2400
Huang J (2000) Asymptotic tracking of a nonminimum phase nonlinear system with nonhyperbolic zero dynamics. IEEE Trans Autom Control 45(3):542–546
Khalil H (2002) Nonlinear systems. Prentice Hall
Acknowledgments
This work is supported by NSFC under Grant No. 61273092.
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© 2016 Springer Science+Business Media Singapore
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Ye, L., Zong, Q., Zhang, X., Wang, D., Dong, Q. (2016). Tracking Control of a Nonminimum Phase Inverted Pendulum. In: Jia, Y., Du, J., Zhang, W., Li, H. (eds) Proceedings of 2016 Chinese Intelligent Systems Conference. CISC 2016. Lecture Notes in Electrical Engineering, vol 405. Springer, Singapore. https://doi.org/10.1007/978-981-10-2335-4_32
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DOI: https://doi.org/10.1007/978-981-10-2335-4_32
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