Neural network model to control an experimental chaotic pendulum.

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 Poincaresections 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!# ,o f 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. @S1063-651X~96!09910-2# 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 attrac- tor 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 sensi- tivity of chaotic systems for small perturbations, this can be achieved using only very small control actions. The possibil- ity to select different kinds of periodic behavior by just se- lecting different UPO's makes this type of control very ap- pealing. 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 experimen- tal systems, we have chosen to extend the method with a neural-network-based process model, and to first test this ex- tension 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. Hubinger et al. @2# controlled a pendulum of which all the state vari- ables were measured. They used the equations of motion to calculate UPO's and to make predictions, and they devel- oped 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. A problem that we envisage when the control strategy is