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Stabilization of a Cart with Inverted Pendulum

  • NONLINEAR SYSTEMS
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Abstract

We consider the problem of stabilizing a cart moving along a straight line with an inverted pendulum installed on it. The control objective is to stabilize the cart at a given target point so that the pendulum is in the upper vertical position. The main difficulty associated with solving this problem is that the two subsystems (the cart and the pendulum) must be stabilized simultaneously using one control. A new control law is proposed based on the introduction of a second-order reference system, the trajectory of which is taken as the target one for the cart with the pendulum. By extending the reference system to the fourth order and introducing an algebraic condition coupling the two systems, the target trajectory is found in the four-dimensional phase space of the original system and a control law is constructed that ensures the trajectory of the closed-loop system asymptotically approaching the target one. The control law obtained in this paper is applicable to systems with an arbitrary ratio of pendulum and cart masses, since the closed-loop system does not depend on the mass characteristics of the system. The range of the system parameters is found for which the linearized system is stable. The presentation is illustrated with numerical examples that demonstrate the efficiency of the proposed control.

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REFERENCES

  1. Neimark, Yu.I., Matematicheskoe modelirovanie kak nauka i iskusstvo (Mathematical Modeling as Science and Art), Nizhny Novgorod: Izd. Nizhegorod. Univ., 2010.

    Google Scholar 

  2. Martynenko, Y.G. and Formal’skii, A.M., Controlled pendulum on a movable base, Mech. Solids, 2013, vol. 48, no. 1, pp. 6–18.

    Article  Google Scholar 

  3. Formal’skii, A.M., Upravlenie dvizheniem neustoichivykh ob”ektov (Motion Control of Unstable Plants), Moscow: Fizmatlit, 2014.

    Google Scholar 

  4. Getz, N.H. and Hedrick, J.K., An internal equilibrium manifold method of tracking for nonlinear nonminimum phase systems, ACC Proc., 1995, pp. 1–5.

  5. Srinivasan, B., Huguenin, P., and Bonvin, D., Global stabilization of an inverted pendulum—Control strategy and experimental verification, Automatica, 2009, vol. 45, pp. 265–269.

    Article  MathSciNet  Google Scholar 

  6. Lee, J., Mukherjee, R., and Khalil, H.K., Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties, Automatica, 2015, vol. 54, no. 4, pp. 146–157.

    Article  MathSciNet  Google Scholar 

  7. Teel, A.R., A nonlinear small gain theorem for the analysis of control systems with saturation, Trans. Autom. Contr. IEEE, 1996, vol. 41, no. 9, pp. 1256–1270.

    Article  MathSciNet  Google Scholar 

  8. Gordillo, F. and Aracil, J., A new controller for the inverted pendulum on a cart, Int. J. Robust Nonlinear Control, 2008, no. 18, pp. 1607–1621.

  9. Magni, L., Scattolini, R., and Aström, K.J., Global stabilization of the inverted pendulum using model predictive control, IFAC Proc., 2002, vol. 35, no. 1, pp. 141–146.

  10. Utkin, V., Guldner, J., and Shi, J., Sliding Mode Control in Electromechanical Systems, CRC Press, 2009.

  11. Matrosov, I.V., Morozov, Yu.V., and Pesterev, A.V., Control of the robot-wheel with a pendulum, Proc. 15th Int. Conf. Stab. Oscillations Nonlinear Control Syst. (Pyatnitskiy’s Conf.) (STAB), IEEE, 2020, pp. 1–4.

  12. Pesterev, A.V., Morozov, Yu.V., and Matrosov, I.V., On optimal selection of coefficients of a controller in the point stabilization problem for a robot-wheel, Commun. Comput. Inf. Sci. (CCIS), 2020, vol. 1340, pp. 236–249.

    Google Scholar 

  13. Sergeev, V.S., On one method for obtaining estimates of domains of attraction using Lyapunov functions constructed by a numerical method, Zh. Vychisl. Mat. Mat. Fiz., 1978, vol. 18, no. 5, pp. 1154–1161.

    MathSciNet  Google Scholar 

  14. Kamenetskii, V.A., Construction of domains of attraction by the method of Lyapunov functions, Avtom. Telemekh., 1994, no. 6, pp. 10–26.

  15. Chiang, H.D. and Thorp, J.S., Stability regions of nonlinear dynamical systems: a constructive methodology, IEEE Trans. Autom. Control, 1989, vol. 30, no. 12, pp. 1229–1241.

    Article  MathSciNet  Google Scholar 

  16. Gantmakher, F.R., Teoriya matrits. 5-e izd. (Theory of Matrices. 5th ed.), Moscow: Fizmatlit, 2010.

    Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    A. V. Pesterev & Yu. V. Morozov

Authors
  1. A. V. Pesterev
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  2. Yu. V. Morozov
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Correspondence to A. V. Pesterev or Yu. V. Morozov.

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Translated by V. Potapchouck

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Pesterev, A.V., Morozov, Y.V. Stabilization of a Cart with Inverted Pendulum. Autom Remote Control 83, 78–91 (2022). https://doi.org/10.1134/S0005117922010064

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  • DOI: https://doi.org/10.1134/S0005117922010064

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