ABSTRACT
The describing function method, abbreviated as DF, was developed in several countries in the 1940s [1], to answer the question: “What are the necessary and sufficient conditions for the nonlinear feedback system of Figure 18.1 to be stable?” The problem still remains unanswered for a system with static nonlinearity, n(x), and linear plant G(s), All of the original investigators found limit cycles in control systems and observed that, in many instances with structures such as Figure 18.1 the waveform of the oscillation at the input to the nonlinearity was almost sinusoidal. If, for example, the nonlinearity in Figure 18.1 is an ideal relay, that is, has an on–off characteristic, so that an odd symmetrical input waveform will produce a square wave at its output, the output of G(s) will be almost sinusoidal when G(s) is a low-pass filter which attenuates the higher harmonics in the square wave much more than the fundamental. It was, therefore, proposed that the nonlinearity should be represented by its gain to a sinusoid and that the conditions for sustaining a sinusoidal limit cycle be evaluated to assess the stability of the feedback loop. This gain of the nonlinearity in response to a sinusoid is a function of the amplitude of the sinusoid and is known as the DF. Because DF methods can be used for other than a single sinusoidal input to distinguish this DF it is also referred to as the single sinusoidal DF or sinusoidal DF.
