Abstract
Chapter 8 devotes to the classical problem of stabilization of the controlled inverted pendulum. The problem of stabilization for the mathematical model of the controlled inverted pendulum during many years is very popular among the researchers. Unlike of the classical way of stabilization in which the stabilized control is a linear combination of the state and velocity of the pendulum here another way of stabilization is proposed. It is supposed that only the trajectory of the pendulum can be observed and stabilized control depends on whole trajectory of the pendulum. Linear and nonlinear models of the controlled inverted pendulum by stochastic perturbations are considered, in particular, under influence of Markovian stochastic perturbations. Via the general method of construction of Lyapunov functionals sufficient conditions for stabilization of zero solution by stochastic perturbations are obtained, nonzero steady-state solutions are investigated. 38 figures show a behavior of the controlled inverted pendulum in the case of stable and unstable equilibrium.
Keywords
- Inverted Pendulum
- Nonzero Steady-state Solution
- Constructing Lyapunov Functionals
- Stochastic Perturbations
- Asymptotic Stability
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References
Acheson DJ (1993) A pendulum theorem. Proc R Soc Lond Ser A, Math Phys Sci 443(1917):239–245
Acheson DJ, Mullin T (1993) Upside-down pendulums. Nature 366(6452):215–216
Blackburn JA, Smith HJT, Gronbech-Jensen N (1992) Stability and Hopf bifurcations in an inverted pendulum. Am J Phys 60(10):903–908
Borne P, Kolmanovskii V, Shaikhet L (1999) Steady-state solutions of nonlinear model of inverted pendulum. Theory Stoch Process 5(21)(3–4):203–209. Proceedings of the third Ukrainian–Scandinavian conference in probability theory and mathematical statistics, 8–12 June 1999, Kyiv, Ukraine
Borne P, Kolmanovskii V, Shaikhet L (2000) Stabilization of inverted pendulum by control with delay. Dyn Syst Appl 9(4):501–514
Imkeller P, Lederer Ch (2001) Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum. Dyn Syst 16:29–61
Kapitza PL (1965) Dynamical stability of a pendulum when its point of suspension vibrates, and pendulum with a vibrating suspension. In: ter Haar D (ed) Collected papers of P.L. Kapitza, vol 2. Pergamon Press, London, pp 714–737
Levi M (1988) Stability of the inverted pendulum—a topological explanation. SIAM Rev 30(4):639–644
Levi M, Weckesser W (1995) Stabilization of the inverted linearized pendulum by high frequency vibrations. SIAM Rev 37(2):219–223
Mata GJ, Pestana E (2004) Effective Hamiltonian and dynamic stability of the inverted pendulum. Eur J Phys 25:717–721
Mitchell R (1972) Stability of the inverted pendulum subjected to almost periodic and stochastic base motion—an application of the method of averaging. Int J Non-Linear Mech 7:101–123
Ovseyevich AI (2006) The stability of an inverted pendulum when there are rapid random oscillations of the suspension point. Int J Appl Math Mech 70:762–768
Sanz-Serna JM (2008) Stabilizing with a hammer. Stoch Dyn 8:47–57
Shaikhet L (2005) Stability of difference analogue of linear mathematical inverted pendulum. Discrete Dyn Nat Soc 2005(3):215–226
Shaikhet L (2009) Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations. Discrete Contin Dyn Syst 24(4):1335–1343. doi:10.3934/dcds.2009.24.1335
Shaikhet L (2011) Lyapunov functionals and stability of stochastic difference equations. Springer, London
Sharp R, Tsai Y-H, Engquist B (2005) Multiple time scale numerical methods for the inverted pendulum problem. In: Multiscale methods in science and engineering. Lecture notes computing science and engineering, vol 44. Springer, Berlin, pp 241–261
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Shaikhet, L. (2013). Stabilization of the Controlled Inverted Pendulum by a Control with Delay. In: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00101-2_8
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DOI: https://doi.org/10.1007/978-3-319-00101-2_8
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