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**Topic under **Control Mode

**Source:** alexandria.tue.nl

**File size:** **201.15 KB**

**File type:** **pdf**

**Last download on:**
Mon Nov 12, 2018 05:58:37 AM

__Short Desciption__:

A feedforward neural network was trained to predict the motion of an experimental, driven, and damped
pendulum operating in a chaotic regime. The network learned the behavior of the pendulum from a time series
of the pendulum’s angle, the single measured variable. The validity of the neural network model was assessed
by comparing Poincare´ sections of measured and model-generated data. The model was used to find unstable
periodic orbits ~UPO’s!, up to period 7. Two selected orbits were stabilized using the semicontinuous control
extension, as described by De Korte, Schouten, and van den Bleek @Phys. Rev. E 52, 3358 ~1995!#, of the
well-known Ott-Grebogi-Yorke chaos control scheme @Phys. Rev. Lett. 64, 1196 ~1990!#. The neural network
was used as an alternative to local linear models. It has two advantages: ~i! it requires much less data, and ~ii!
it can find many more UPO’s than those found directly from the measured time series.

__Summary__:

In the last ten years, many physical systems that exhibit
seemingly random behavior have been demonstrated to be
low-dimensional chaotic. In practice, the presence of chaos is
often undesirable, and insights from nonlinear dynamics
have been used to account for chaos in process design. At the
same time, Ott, Grebogi, and Yorke ~OGY! @1# developed a
control scheme that can be used to exploit chaotic behavior.
The OGY method is based on the observation that the attractor
of a chaotic system typically contains an infinite number
of unstable periodic orbits ~UPO’s!. All that is needed to
change the system behavior from chaotic to periodic is to
select one of these UPO’s and stabilize it. Due to the sensitivity
of chaotic systems for small perturbations, this can be
achieved using only very small control actions. The possibility
to select different kinds of periodic behavior by just selecting
different UPO’s makes this type of control very appealing.
Thus far, the OGY method has been applied to simple
chaotic systems. To develop the chaos control methodology
further, and make it applicable to more complex experimental
systems, we have chosen to extend the method with a
neural-network-based process model, and to first test this extension
on a comprehensive experimental system, a driven
and damped pendulum. For control, one needs ~i! a method
to find UPO’s, and ~ii! a model that can predict future states
of the system from the present, measured state. Hu¨binger
et al. @2# controlled a pendulum of which all the state variables
were measured. They used the equations of motion to
calculate UPO’s and to make predictions, and they developed
a semicontinuous control ~SCC! extension of the OGY
method to cope with the large unstable eigenvalues of the
stabilized UPOs. De Korte, Schouten, and van den Bleek @3#
controlled a different, less ideal pendulum of which only one
state variable, its angle, was measured. Delayed values of the
angle were used to obtain a complete representation of the
state. UPO’s were found by searching the measured data for
close returning points, and the predictions were made by
local linear models, fitted directly to the measured data.

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"Neural network model to control an experimental chaotic pendulum"

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