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Hysteretic Nonlinearity in Inverted Pendulum Problem

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Structural Nonlinear Dynamics and Diagnosis

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 168))

Abstract

This work is dedicated to the problem of inverted pendulum under hysteretic nonlinearity in the form of backlash in the suspension point. We present the results for various motion of the suspension point, namely, the vertical and horizontal motions. We consider the mathematical model of inverted pendulum with vertically oscillating suspension and in the frame of presented model the explicit stability criteria for the linearized equations of motion are found. Dependencies between initial conditions and driven parameters, that provide periodic oscillations of the pendulum, are obtained. In the next step we consider the mathematical model of inverted pendulum under state feedback control (horizontal motion of suspension). Analytic results for the stability criteria as well as for the solution of linearized equation are observed and analyzed. The theorems that determine stabilization of the considered system are formulated and discussed together with the question on the optimal control. We also investigate the elastic inverted pendulum with backlash in the suspension point (horizontal motion). The problem of stabilization together with an optimization problem for such a system is considered. Algorithm (based on the bionic model) which provides the effective procedure for finding of optimal parameters is presented and applied to considered system. Phase portraits and dynamics of the Lyapunov function are also presented and discussed.

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Notes

  1. 1.

    The one-dimensional swinging inverted pendulum with two degrees of freedom is a popular demonstration of using feedback control to stabilize an open-loop unstable system. Since the system is inherently nonlinear, it has been using extensively by the control engineers to verify a modern control theory. In this system, an inverted pendulum is attached to a cart equipped with a motor that drives it along a horizontal track [14].

  2. 2.

    Here we would like to note that in three considered cases we introduce the mathematical description of backlash in the ways that are comfortable to use in the concrete case. However all these descriptions are based on the operator technique with small variations that are presented in the corresponding sections.

  3. 3.

    Here we would like to note that both of the cylinder and piston are ideal, absolutely rigid and can move along the y-axis in the infinite ranges.

  4. 4.

    It should be pointed out that such a periodic behavior of the piston’s acceleration (i.e., the fact that the acceleration of the piston changes from \(-a\omega ^{2}\) to \(a\omega ^{2}\)) is an assumption of the model presented in this paper. Such a model allows us to obtain some analytical results (the explicit conditions for the stability zones). Also, the numerical simulations are most effectively in the frame of this model. Moreover, such a model of the piston’s behavior most effectively and adequately describes the dynamics of the parts of real technical devices.

  5. 5.

    Here we use the following notations: \(a_{x}=\frac{\partial a}{\partial x}\), \(a_{t}=\frac{\partial a}{\partial t}\).

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Acknowledgments

This work is supported by the RFBR grant 13-08-00532-a.

Author information

Authors and Affiliations

  1. Meteorology Department, Zhukovsky–Gagarin Air Force Academy, Starykh Bolshevikov St. 54 “A”, 394064, Voronezh, Russia

    Mikhail E. Semenov

  2. Digital Technologies Department, Voronezh State University, Universitetskaya Sq. 1, 394006, Voronezh, Russia

    Mikhail E. Semenov, Peter A. Meleshenko & Andrey M. Solovyov

  3. Mathematics Department, Voronezh State University of Architecture and Civil Engineering, XX-letiya Oktyabrya St. 84, 394006, Voronezh, Russia

    Mikhail E. Semenov

  4. Communication Department, Zhukovsky–Gagarin Air Force Academy, Starykh Bolshevikov St. 54 “A”, 394064, Voronezh, Russia

    Peter A. Meleshenko

  5. Materials Science and Nanosystems Technologies Department, Voronezh State University, Universitetskaya Sq. 1, 394006, Voronezh, Russia

    Andrey M. Semenov

Authors
  1. Mikhail E. Semenov
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  2. Peter A. Meleshenko
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Correspondence to Mikhail E. Semenov .

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Editors and Affiliations

  1. Laboratory of Mechanics, Hassan II-Casablanca University, Casablanca, Morocco

    Mohamed Belhaq

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Semenov, M.E., Meleshenko, P.A., Solovyov, A.M., Semenov, A.M. (2015). Hysteretic Nonlinearity in Inverted Pendulum Problem. In: Belhaq, M. (eds) Structural Nonlinear Dynamics and Diagnosis. Springer Proceedings in Physics, vol 168. Springer, Cham. https://doi.org/10.1007/978-3-319-19851-4_22

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