Designing autonomous relay systems with chaotic motion

An investigation has been conducted, using simulation, of the existence of chaotic motion in several relay feedback systems. Particular emphasis has been placed on determining when chaotic motion might exist from a knowledge of the unstable limit cycles predicted by the Tsypkin method, the largest amplitude unstable limit cycle being approximately sinusoidal. It is shown that this sinusoidal limit cycle can be calculated quite accurately by the describing function method, and it is found essentially to bound the region of chaotic motion. The chaotic motion gives the appearance of jumps between two or more of the unstable limit cycles found within the region. These unstable limit cycles exhibit two or more oscillations per half period, or spirals if viewed on a phase plane, and their peak amplitudes can be predicted approximately from consideration of the relay switching levels and the DC gain of the linear transfer function.<>