Research Papers

Control-Based Continuation of Unstable Periodic Orbits

[+] Author and Article Information
Jan Sieber1

Department of Mathematics, University of Portsmouth, PO1 3HF Portsmouth, UKjan.sieber@port.ac.uk

Bernd Krauskopf

Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK

David Wagg, Simon Neild

Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK

Alicia Gonzalez-Buelga

Department of Civil Engineering, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain


Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011005 (Sep 28, 2010) (9 pages) doi:10.1115/1.4002101 History: Received April 30, 2009; Revised December 11, 2009; Published September 28, 2010; Online September 28, 2010

We present an experimental procedure to track periodic orbits through a fold (saddle-node) bifurcation and demonstrate it with a parametrically excited pendulum experiment where the tracking parameter is the amplitude of the excitation. Specifically, we track the initially stable period-one rotation of the pendulum through its fold bifurcation and along the unstable branch. The fold bifurcation itself corresponds to the minimal amplitude that supports sustained rotation. Our scheme is based on a modification of time-delayed feedback in a continuation setting and we show for an idealized model that it converges with the same efficiency as classical proportional-plus-derivative control.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 2

Loss of stability in the controlled system in the (ω,p)-plane for varying control gains G. The gains G vary from 0 to 34 in steps of 2. The thin dark curves show when systems 3,4, when using classical PD control with the control target ϕ0=ϕ∗, loses stability. The thick light curves show when systems 3,7 loses stability. Fixed parameters are as estimated for the physical pendulum discussed in Sec. 4; see Table 1.

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Figure 7

Panel (a) shows a short time window of the signal ϕ(t)−ϕ0(t), entering the feedback loop (frequency ω=4 Hz). Open circles indicate that the signal ϕ−ϕ0 has been accepted as periodic (the map M1 is evaluated from the forcing interval before each open circle); the filled circles correspond to accepted fixed points of M1 (and, thus, to points on the curve shown in Fig. 4). The panels (b) and (c) show how the parameters p and ϕ̃0 have been varied by the iteration 29.

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Figure 1

Loss of stability for the controlled system at frequency ω=3 Hz and gain G1=1 ms−2 and G2=0.5 ms−1 in Eqs. 3,4,7. Note that the uncontrolled pendulum 2 loses its stability already at the fold point. The fixed parameters were set to values corresponding to the experimental setup discussed in Sec. 4 (see Table 1).

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Figure 3

Experimental setup: (a) sketch and (b) photo showing the pendulum and the actuator. The dSpace board obtains ϕ=θ−ωt and outputs avg[ϕ] and yr by computing recursion 7, the ODEs 22,27, and the reset law 26 in real-time. The inputs p and ϕ̃0 are piecewise constant and obtained by recursion 29. See Fig. 7 for a typical time profile of inputs and outputs.

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Figure 4

Experimental continuation for frequency ω=4 Hz. The y-axis shows the average of ϕ, which corresponds to the phase of the pendulum relative to the pivot motion.

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Figure 5

Time profiles ϕ(t) of recorded periodic rotations. Each profile corresponds to one circle in Fig. 4 (only every second solution is plotted); see Table 1 for parameters.

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Figure 6

Overview of performed continuation runs in the forcing amplitude p for frequencies ranging from right to left between 2 Hz and 5 Hz. Continuations at smaller frequencies have a fold at higher amplitudes.




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