Abstract
The set of states (controllability domain) from which an unstable object can be steered to a desired operational regime is bounded in the phase space, if the control resources are restricted. Under admissible (with given resources) feedback control, the basin of attraction of the desired regime belongs to this controllability domain. The problem of control design to maximize the basin of attraction is discussed in this paper. Several systems with underactuation degree one are studied. A system of gyroscopic stabilization of the unstable upright position of a two-wheel bicycle is described also. An active field of research exists, due to the applications of underactuated systems.
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References
Formalskii, A.M.: Stabilization of an inverted pendulum with a fixed or movable suspension point. Dokl. Math. 73(1), 152–156 (2006)
Formalskii, A.M.: An inverted pendulum on a fixed and a moving base. J. Appl. Math. Mech. 70(1), 56–64 (2006)
Formalskii, A.M.: Controllability and Stability of Systems with Restricted Control Resources. Nauka, Moscow (1974) (in Russian)
Spong, M.W., Corke, P., Lozano, R.: Nonlinear control of the inertia wheel pendulum. Automatica 37, 1845–1851 (2001)
Beznos, A.V., Grishin, A.A., Lenskii, A.V., Okhotsimsky, D.E., Formalskii, A.M.: A pendulum controlled by a flywheel. Dokl. Math. 68(2), 302–307 (2003)
Beznos, A.V., Grishin, A.A., Lenskii, A.V., Okhotsimsky, D.E., Formalskii, A.M.: A flywheel use-based control for a pendulum with fixed suspension point. J. Comput. Syst. Sci. Int. 43(1), 22–33 (2003)
Gorinevsky, D.M., Formalskii, A.M., Schneider, A.Yu.: Force Control of Robotics Systems. CRC, Boca Raton (1997)
Formalskii, A.M.: Global stabilization of a double inverted pendulum with control at the hinge between the links. Mech. Solids 43(5), 687–697 (2008)
Spong, M.W.: The swing up control problem for the acrobot. IEEE Control Syst. Mag. 14(1), 49–55 (1995)
Chernousko, F.L., Ananevskii, I.M., Reshmin, S.A.: Methods of Control of Nonlinear Mechanical Systems. Fizmatlit, Moscow (2006) (in Russian)
Reshmin, S.A.: Method of decomposition in the problem of control of inverted double-link pendulum using one control torque. J. Comput. Syst. Sci. Int. 6, 28–45 (2005)
Chetaev, N.G.: The Stability of Motion. Pergamon, Elmsford (1961)
Chetaev, N.G.: Theoretical Mechanics. Springer, Berlin (1989)
Hu, T., Lin, Z., Qiu, L.: Stabilization of exponentially unstable linear systems with saturating actuators. IEEE Trans. Automat. Control 46(6), 973–979 (2001)
Magnus, K.: Gyroscope. Theorie und Anwendungen. Springer, Heidelberg (1971)
Lenskii, A.V., Formalskii, A.M.: Gyroscopic stabilization of a two-wheeled robot bicycle. Dokl. Math. 70(3), 993–997 (2004)
Lenskii, A.V., Formalskii, A.M.: Two-wheel robot-bicycle with a gyroscopic stabilizer. J. Comput. Syst. Sci. Int. 42(3), 482–489 (2003)
Martynenko, Yu.G., Formalskii, A.M.: The theory of the control of a monocycle. J. Appl. Math. Mech. 69(4), 516–528 (2005)
Aoustin, Y., Formalskii, A.M.: Ball on a beam: stabilization under saturated input control with large basin of attraction. Multibody Syst. Dyn. 21(5), 71–89 (2008)
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Communicated by F.L. Chernousko.
This work has been carried out with financial support from the Russian Foundation for Basic Research, Grants 07-01-92167, 09-01-00593-a.
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Formalskii, A.M. Stabilization of Unstable Mechanical Systems. J Optim Theory Appl 144, 227–253 (2010). https://doi.org/10.1007/s10957-009-9600-x
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DOI: https://doi.org/10.1007/s10957-009-9600-x