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This paper proposes an innovative framework of a parameter estimation procedure based on the well-established relay-feedback experiment paradigm. The novelty consists in consideration of asymmetric dynamics and non-equal static gains of the identified system. A different system behavior after changing the input variable polarity near the operating point is rarely considered or even omitted within relay-based parameter identification tests, in contrast to the common use of asymmetry in the nonlinear relay element. The thing is that many existing relay-based identification techniques in the frequency domain use integrations, assuming that the system output operating point coincides with the setpoint value (i.e., the offset between them is zero). However, this is not true for asymmetric dynamic systems, which yields considerably erroneous parameter estimation as the integration result is highly sensitive to the baseline value. The resulting iterative numerical optimization-based algorithm is built-up using a chain of natural assumptions and step-by-step thought experiments. The proposed framework is applied to the well-established exponential decaying method in this paper. Some computation aspects of the algorithm are discussed. A comparative numerical study illustrates the efficacy of the proposed strategy, where several frequency-fitting-based and descriptive-function-based competitive approaches are considered.