Frequency domain criteria for the absence of zero-input limit cycles in nonlinear discrete-time systems, with applications to digital filters

On the basis of a theorem of Barkin, a number of criteria are derived which give sufficient conditions for the absence of zero-input limit cycles in discrete-time and especially digital systems. These criteria are formulated in the frequency domain and provide a possibility of investigating the absence of a zero-input limit cycle of a specific length
N
. Depending on the particular characteristics of the nonlinearities and the number of nonlinearities in the system, different criteria are obtained. Application of the criteria results in most cases in a linear programming problem. The solutions to this problem for the case of a second-order digital filter with one and with two quantization nonlinearities are discussed.