Abstract
Control algorithms that exploit chaotic behavior and its precursors can vastly improve the performance of many practical and useful systems. Phase-locked loops, for example, are normally designed using linearization. This approximation hides the global dynamics that lead to lock and capture range limits. Design techniques that are equipped to exploit the real nonlinear and chaotic nature of the device can loosen these limitations. The program Perfect Moment is built around a collection of such techniques. Given a differential equation, a control parameter, and two state-space points, the program explores the system's behavior, automatically choosing interesting and useful parameter values and constructing state-space portraits at each one. It then chooses a set of trajectory segments from those portraits, uses them to construct a composite path between the objectives, and finally causes the system to follow that path by switching the parameter value at the segment junctions. Rules embodying theorems and definitions from nonlinear dynamics are used to limit computational complexity by identifying areas of interest and directing and focusing the mapping and search on these areas. Even so, these processes are computationally intensive. However, the sensitivity of a chaotic system's state-space topology to the parameters of its equations and the sensitivity of the paths of its trajectories to state perturbations make this approach rewarding in spite of its computational demands. Reference trajectories found by this design tool exhibit a variety of interesting and useful properties. Perfect Moment balances an inverted pendulum by ``pumping'' the device up, over half a dozen cycles, using roughly one-sixth the torque that a traditional linear controller would exert in this task. In another example, the program uses a detour through a chaotic zone to stabilize a system about 300 times faster than would happen by waiting long enough for the system to reach a regime where traditional control methods obtain. Chaotic zones can also be used to steer trajectories across boundaries of basins of attraction, effectively altering the geometry and the convergence properties of the system's stability regions. An externally-induced chaotic attractor that overlaps the phase-locked loop's original lock range is demonstrated here; the controller designed by the program uses this feature to extend the capture range out to the original lock range limits, allowing the circuit to acquire lock over a wider range of input frequencies than would otherwise be possible.
Full paper here.
This paper is compressed because the original postscript file is immense. To uncompress it, use the unix command gzip or the MacOS/Windows/DOS command StuffIt.