Experimental bifurcation control of a parametric pendulum

Siam Journal on Applied Dynamical Systems, 2005

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos. . S.C. and K.M. acknowledge the support of NSERC through the individual research grants program. M.L. acknowledges the support of NSERC through the USRA program and through the grants of S.C. and K.M. C.A. acknowledges the support of NSERC through the USRA program and of the Department of Applied Mathematics, University of Waterloo.

Chaos Solitons & Fractals, 2009

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge-Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.

Synthesis Lectures on Mechanical Engineering, 2019

Synthesis Lectures on Mechanical Engineering series publishes 60-150 page publications pertaining to this diverse discipline of mechanical engineering. The series presents Lectures written for an audience of researchers, industry engineers, undergraduate and graduate students. Additional Synthesis series will be developed covering key areas within mechanical engineering.

2003

In this chapter the mathematical tools from bifurcation theory are used within the framework of feedback control systems. The first part deals with a simple example where the amplitude of limit cycles and the appearance of period-doubling bifurcations are controlled using a method derived from the frequency domain approach. In the second part, bifurcation theory is used to analyze the dynamical behavior of an inverted pendulum with saturated control. The main objective is to find appropriate values of the controller parameters to achieve the stabilization of the pendulum at the inverted position and, at the same time, to obtain the largest basin of attraction.

… Proceedings, 24–29 …, 2011

Summary. We present a software toolbox that allows to apply continuation methods directly to a controlled lab experiment. This toolbox enables us to systematically explore how stable and unstable steady state periodic vibrations depend on parameters. The toolbox is ...

International journal of …, 2001

The control of nonlinear systems exhibiting bifurcation phenomena has been the subject of active research in recent years. Contrary to regulation or tracking objectives common in classic control, in some applications it is desirable to achieve an oscillatory behavior. Towards this end, bifurcation control aims at designing a controller to modify the bifurcative dynamical behavior of a complex nonlinear system. Among the available methods, the so-called \anti-control" of Hopf bifurcations is one approach to design limit cycles in a system via feedback control. In this paper, this technique is applied to obtain oscillations of prescribed amplitude in a simple mechanical system: an underactuated pendulum. Two different nonlinear control laws are described and analyzed. Both are designed to modify the coefficients of the linearization matrix of the system via feedback. The fi rst law modifi es those coefficients that correspond to the physical parameters, whereas the second one changes some null coefficients of the linearization matrix. The latter results in a simpler controller that requires the measurement of only one state of the system. The dependence of the amplitudes as function of the feedback gains is obtained analytically by means of local approximations, and over a larger range by numerical continuation of the periodic solutions. Theoretical results are contrasted by both computer simulations and experimental results.

Automatica, 1995

nonlinear systems about a periodic orbit.

HAL (Le Centre pour la Communication Scientifique Directe), 2000

Experimental and numerical investigations are carried out on an autoparametric system consisting of a composite pendulum attached to a harmonically base excited mass-spring subsystem. The dynamic behavior of such a mechanical system is governed by a set of coupled nonlinear equations with periodic parameters. Particular attention is paid to the dynamic behavior of the pendulum. The periodic doubling bifurcation of the pendulum is determined from the semi-trivial solution of the linearized equations using two methods: a trigonometric approximation of the solution and a symbolic computation of the Floquet transition matrix based on Chebyshev polynominal expansions. The set of nonlinear differential equations is also integrated with respect to time using a finite difference scheme and the motion of the pendulum is analyzed via phase-plane portraits and Poincare maps. The predicted results are experimentally validated through an experimental set-up equipped with an optoelectronic set sensor that is used to measure the angular displacement of the pendulum. Period doubling and chaotic motions are observed.

International Journal of Bifurcation and Chaos, 2007

The dynamical behavior of a simple pendulum hanging from a rotating arm has been investigated. The system is invariant under rotations around the axis and can be formulated as a two-degrees of freedom integrable Hamiltonian system in the absence of external forcing. The bifurcation diagram is organized around the relative equilibria (solutions that are invariant under the symmetry) and bridges connecting different bifurcation points. Special attention has been given to those solutions that could shed some light into the stabilization of the upside down solution and the control problem.

2009

A model of a multidimensional system, consisting of a chain of nonlinearly coupled chaotic pendula subjected to a chaos-inducing harmonic parametric excitation, is introduced. The nonlinear dynamics of the chain is analyzed with attention to the synchronization phenomena. The control of chaos (suppression and enhancement) through the introduction of a second periodic excitation is investigated, by a numerical approach (Lyapunov exponents), firstly with reference to the simple case of a single pendulum; then the effectiveness of the method is studied on the chain, by applying the localized control on a minimal number of pendula.

Communications in Nonlinear Science and Numerical Simulation, 2019

The nonlinear dynamics of a parametrically excited pendulum is addressed. The proposed analytical approach aims at describing the pendulum dynamics beyond the simplified regimes usually considered in literature, where stationarity and small amplitude oscillations are assumed. Thus, by combining complexification and Limiting Phase Trajectory (LPT) concepts, both stationary and non-stationary dynamic regimes are considered in the neighborhood of the main parametric resonance, without any restriction on the pendulum oscillation amplitudes. The advantage of the proposed approach lies in the possibility of identifying the strongly modulated regimes for arbitrary initial conditions and high-amplitude excitation, cases in which the conventionally used quasilinear approximation is not valid. The identification of the bifurcations of the stationary states as well as the large-amplitude corrections of the stability thresholds emanating from the main parametric resonance are also provided.

Chaos, Solitons & Fractals, 2004

Chaotic behavior of dynamical systems offers a rich variety of orbits, which can be controlled by small perturbations in either a specific parameter of the system or a dynamical variable. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. At first, it is considered signals generated by numerical integration of the mathematical model. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method is employed with this aim. Finally, an analysis related to the effect of noise in controlling chaos is of concern. Results show situations where these techniques may be used to control chaos in mechanical systems.

Science & Education, 2004

This paper conveys information about a Physics laboratory experiment for students with some theoretical knowledge about oscillatory motion. Students construct a simple pendulum that behaves as an ideal one, and analyze model assumption incidence on its period. The following aspects are quantitatively analyzed: vanishing friction, small amplitude, not extensible string, point mass of the body, and vanishing mass of the string.

IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, 2012

In this work the dynamics and synchronization of the coupled parametric pendulums system is examined with a view to its application for wave energy extraction. The system consisting of two parametric pendulums on a common support has been modeled and its response studied numerically, with a main focus on synchronized rotation. Different methods of controlling the response the pendulums have been introduced and compared. Numerical results have been verified in experimental studies.