×

zbMATH — the first resource for mathematics

On topological obstructions to global stabilization of an inverted pendulum. (English) Zbl 1386.93259
Summary: We consider a classical problem of control of an inverted pendulum by means of a horizontal motion of its pivot point. We suppose that the control law can be non-autonomous and non-periodic w.r.t. the position of the pendulum. It is shown that global stabilization of the vertical upward position of the pendulum cannot be obtained for any Lipschitz control law, provided some natural assumptions. Moreover, we show that there always exists a solution separated from the vertical position and along which the pendulum never becomes horizontal. Hence, we also prove that global stabilization cannot be obtained in the system where the pendulum can impact the horizontal plane (for any mechanical model of impact). Similar results are presented for several analogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum with an additional torque control.

MSC:
93D21 Adaptive or robust stabilization
70Q05 Control of mechanical systems
93C15 Control/observation systems governed by ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Polekhin, I. Y., Examples of topological approach to the problem of inverted pendulum with moving pivot point, Nelineinaya Dinamika [Russian J. Nonlinear Dyn.], 10, 4, 465-472, (2014) · Zbl 1353.70051
[2] Polekhin, I., Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Anal. TMA, 128, 100-105, (2015) · Zbl 1372.70023
[3] Mori, S.; Nishihara, H.; Furuta, K., Control of unstable mechanical system control of pendulum, Internat. J. Control, 23, 5, 673-692, (1976)
[4] Henders, M.; Soudack, A., “in-the-large” behaviour of an inverted pendulum with linear stabilization, Int. J. Non-Linear Mech., 27, 1, 129-138, (1992) · Zbl 0761.70025
[5] Lin, C. E.; Sheu, Y.-R., A hybrid-control approach for pendulum-car control, IEEE Trans. Ind. Electron., 39, 3, 208-214, (1992)
[6] Chung, C. C.; Hauser, J., Nonlinear control of a swinging pendulum, Automatica, 31, 6, 851-862, (1995) · Zbl 0825.93535
[7] Ludvigsen, H.; Shiriaev, A.; Egeland, O., Stabilization of stable manifold of upright position of the spherical pendulum, (American Control Conference, 1999. Proceedings of the 1999, Vol. 6, (1999), IEEE), 4034-4038
[8] Shiriaev, A.; Ludvigsen, H.; Egeland, O.; Pogromsky, A., On global properties of passivity based control of the inverted pendulum, (Proceedings of the 38th IEEE Conference on Decision and Control, 1999, Vol. 3, (1999), IEEE), 2513-2518
[9] Shiriaev, A.; Ludvigsen, H.; Egeland, O., Swinging up of the spherical pendulum, IFAC Proc. Vol., 32, 2, 2193-2198, (1999)
[10] Åström, K. J.; Furuta, K., Swinging up a pendulum by energy control, Automatica, 36, 2, 287-295, (2000) · Zbl 0941.93543
[11] Angeli, D., Almost global stabilization of the inverted pendulum via continuous state feedback, Automatica, 37, 7, 1103-1108, (2001) · Zbl 0980.93064
[12] Zhao, J.; Spong, M. W., Hybrid control for global stabilization of the cart-pendulum system, Automatica, 37, 12, 1941-1951, (2001) · Zbl 1005.93041
[13] Spong, M. W.; Corke, P.; Lozano, R., Nonlinear control of the reaction wheel pendulum, Automatica, 37, 11, 1845-1851, (2001) · Zbl 0978.93535
[14] Shiriaev, A. S.; Ludvigsen, H.; Egeland, O., Swinging up the spherical pendulum via stabilization of its first integrals, Automatica, 40, 1, 73-85, (2004) · Zbl 1037.70015
[15] Chatterjee, D.; Patra, A.; Joglekar, H. K., Swing-up and stabilization of a cart-pendulum system under restricted cart track length, Systems Control Lett., 47, 4, 355-364, (2002) · Zbl 1106.70332
[16] Gordillo, F.; Aracil, J., A new controller for the inverted pendulum on a cart, Internat. J. Robust Nonlinear Control, 18, 17, 1607-1621, (2008) · Zbl 1160.93378
[17] Sugihara, T.; Nakamura, Y.; Inoue, H., Real-time humanoid motion generation through ZMP manipulation based on inverted pendulum control, (IEEE International Conference on Robotics and Automation, 2002. Proceedings, Vol. 2, ICRA’02, (2002), IEEE), 1404-1409
[18] Furuta, K., Control of pendulum: from super mechano-system to human adaptive mechatronics, (42nd IEEE Conference on Decision and Control, 2003. Proceedings, Vol. 2, (2003), IEEE), 1498-1507
[19] Kajita, S.; Kanehiro, F.; Kaneko, K.; Fujiwara, K.; Harada, K.; Yokoi, K.; Hirukawa, H., Resolved momentum control: humanoid motion planning based on the linear and angular momentum, (Proceedings. 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2003, Vol. 2, (IROS 2003), (2003), IEEE), 1644-1650
[20] Popovic, M.; Hofmann, A.; Herr, H., Angular momentum regulation during human walking: biomechanics and control, (2004 IEEE International Conference on Robotics and Automation, 2004. Proceedings, Vol. 3, ICRA’04, (2004), IEEE), 2405-2411
[21] Kajita, S.; Morisawa, M.; Harada, K.; Kaneko, K.; Kanehiro, F.; Fujiwara, K.; Hirukawa, H., Biped walking pattern generator allowing auxiliary ZMP control, (2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, (2006), IEEE), 2993-2999
[22] Bhat, S. P.; Bernstein, D. S., A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, Systems Control Lett., 39, 1, 63-70, (2000) · Zbl 0986.93063
[23] Kozlov, V. V., Integrability and non-integrability in Hamiltonian mechanics, Russian Math. Surveys, 38, 1, 1-76, (1983) · Zbl 0525.70023
[24] Auckly, D.; Kapitanski, L., Mathematical problems in the control of underactuated systems, (Signal, I. M.; Sulem, C., Nonlinear Dynamics and Renormalization Groupa, (2001)), 29-40, 27 · Zbl 0982.93059
[25] Ważewski, T., Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20, 279-313, (1947) · Zbl 0032.35001
[26] Reissig, R.; Sansone, G.; Conti, R., Qualitative Theorie Nichtlinearer Differentialgleichungen, (1963), Edizioni Cremonese · Zbl 0114.04302
[27] Polekhin, I., On forced oscillations in groups of interacting nonlinear systems, Nonlinear Anal. TMA, 135, 120-128, (2016) · Zbl 1353.70050
[28] Bolotin, S. V.; Kozlov, V. V., Calculus of variations in the large, existence of trajectories in a domain with boundary, and whitney’s inverted pendulum problem, Izv.: Math., 79, 5, 894, (2015) · Zbl 1367.37053
[29] Demidovich, B., Lectures on the Mathematical Theory of Stability, (1967), Nauka Moscow · Zbl 0155.41601
[30] Wilson, F. W., The structure of the level surfaces of a Lyapunov function, J. Differential Equations, 3, 3, 323-329, (1967) · Zbl 0152.28701
[31] Srzednicki, R., Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. TMA, 22, 6, 707-737, (1994) · Zbl 0801.34041
[32] Srzednicki, R.; Wójcik, K.; Zgliczyński, P., Fixed point results based on the ważewski method, (Handbook of Topological Fixed Point Theory, (2005), Springer), 905-943 · Zbl 1079.37012
[33] Krasovskii, N. N., Stability of Motion, (1963), Stanford University Press Stanford · Zbl 0109.06001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.