Limit cycles in high-resolution quantized feedback systems

In this paper, the existence of limit cycles in high-resolution quantized feedback systems is examined. It is well known that the relay and the quantized feedback systems exhibit self-oscillations, due to their switching nature. However, the quantizer is a more general nonlinearity as compared to the relay, due to its switching at multiple discrete levels. An extension of periodic switching conditions uncovers the existence of self-oscillations in some systems under high quantization resolution. Multiple limit cycle solutions of switching instants and periods have been found, depending on the initial states of the system. Further analysis on the stability of the limit cycle via the Jacobian of the Poincare map reveals numerical bounds on the quantization step size for a stable limit cycle. Analytical results on the existence of limit cycles in first and second order systems are also presented.