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Control of an Inverted Pendulum on a Wheel

  • NONLINEAR SYSTEMS
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Abstract

We consider a mechanical system consisting of a wheel and a pendulum suspended on the wheel axis. The wheel rolls on a horizontal surface. The problem of simultaneous stabilization of the vertical position of the pendulum and a given position of the wheel is considered. A well-known difficulty associated with this problem is that the use of a single control serves two purposes—to stabilize the pendulum angle and the wheel rotation angle. The application of the output feedback linearization method, with the pendulum angle chosen as the output, leads to the appearance of unstable zero dynamics in the closed-loop system.

It is shown that if we take the sum of the pendulum angle and the wheel rotation angle as the output of the system, then the zero dynamics of the closed-loop system turns out to be stable, albeit not asymptotically. A method for asymptotic stabilization of the equilibrium of the closed-loop system is proposed, and an estimate for the attraction basin is constructed. The construction of the estimate is reduced to a problem on the solvability of linear matrix inequalities.

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REFERENCES

  1. Martynenko, Yu.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 

  2. Formal’skii, A.M., Upravlenie dvizheniem neustoichivykh ob”ektov (Control of Motion of Unstable Objects), Moscow: Fizmatlit, 2012.

    Google Scholar 

  3. Khalil, Kh.K., Nelineinye sistemy (Nonlinear Systems), Moscow–Izhevsk: IKI-RKhD, 2009.

    Google Scholar 

  4. Tkachev, S.B., Stabilization of nonminimum phase affine systems using linearization in part of the variables, Nauka Obraz. Izd. MGTU im. N.E. Baumana, 2011, no. 11, pp. 1–29.

  5. Utkin, V.I., Guldner, J., and Shi, J., Sliding Mode Control in Electro-Mechanical Systems, Boca Raton: CRC Press, 2009.

    Google Scholar 

  6. Jung-Su Ha and Ju-Jang Lee, Position control of mobile two wheeled inverted pendulum robot by sliding mode control, Proc. 12th Int. Conf. Control Autom. Syst., 2012, pp. 715–719.

  7. Zhijun Li, Chenguang Yang, and Liping Fan, Advanced Control of Wheeled Inverted Pendulum Systems, Berlin–Heidelberg–New York: Springer, 2013.

    Google Scholar 

  8. Pesterev, A.V. and Morozov, Yu.V., Stabilization of a cart with inverted pendulum, Autom. Remote Control, 2022, vol. 83, no. 1, pp. 78–91.

    Article  MathSciNet  Google Scholar 

  9. 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.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Reshmin, S.A. and Chernous’ko, F.L., A time-optimal control synthesis for a nonlinear pendulum, J. Comput. Syst. Sci. Int., 2007, vol. 46, no. 1, pp. 9–18.

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. 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.

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

    Google Scholar 

  15. https://wxMaxima-developers.github.io/wxmaxima/ .

  16. Polyak, B.T., Khlebnikov, M.V., and Rapoport, L.B., Matematicheskaya teoriya avtomaticheskogo upravleniya (Mathematical Theory of Automatic Control), Moscow: URSS, 2019.

    Google Scholar 

  17. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1983.

    MATH  Google Scholar 

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Authors and Affiliations

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

    L. B Rapoport & A. A. Generalov

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  1. L. B Rapoport
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  2. A. A. Generalov
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Correspondence to L. B Rapoport or A. A. Generalov.

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Translated by V. Potapchouck

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Rapoport, L.B., Generalov, A.A. Control of an Inverted Pendulum on a Wheel. Autom Remote Control 83, 1151–1171 (2022). https://doi.org/10.1134/S000511792208001X

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  • DOI: https://doi.org/10.1134/S000511792208001X

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