We use a saturation function with memory instead of a pure saturation function to generate a limit cycle in order to find one point information of a plant in the frequency domain. The saturation function with memory is useful in the presence of noise and/or a short duration of short duration of external disturbances. We analyze the error caused by the approximation that the saturation function with memory treated as a pure saturation function. We propose a new tuning formula for PID controller which can be applied a saturation function having memory with an arbitrary memory size. We show that the proposed method is more accurate than that of the approximation method via an example.