For nonlinear systems consisting of a feedback connection between a linear system and a static nonlinearity, it is possible to obtain a generalization of Nyquist's stability criterion based on the idea of describing functions. The basic concept involves approximating the response of a static nonlinearity to an input u = A e^{j \omega t} as an output y = N(A) (A e^{j \omega t}), where N(A) \in \mathbb{C} represents the (amplitude-dependent) gain and phase associated with the nonlinearity.

Stability analysis of a linear system H(s) with a feedback nonlinearity F(x) is done by looking for amplitudes A and frequencies \omega such that

H(j\omega) N(A) = -1

If such an intersection exists, it indicates that there may be a limit cycle of amplitude A with frequency \omega.

Describing function analysis is a simple method, but it is approximate because it assumes that higher harmonics can be neglected.

The function :func:`~control.describing_function` can be used to compute the describing function of a nonlinear function:

N = ct.describing_function(F, A)

Stability analysis using describing functions is done by looking for amplitudes a and frequencies :math`omega` such that

H(j\omega) = \frac{-1}{N(A)}

These points can be determined by generating a Nyquist plot in which the transfer function H(j\omega) intersections the negative reciprocal of the describing function N(A). The :func:`~control.describing_function_response` function computes the amplitude and frequency of any points of intersection:

response = ct.describing_function_response(H, F, amp_range[, omega_range])
response.intersections      # frequency, amplitude pairs

A Nyquist plot showing the describing function and the intersections with the Nyquist curve can be generated using response.plot(), which calls the :func:`~control.describing_function_plot` function.

To facilitate the use of common describing functions, the following nonlinearity constructors are predefined:

friction_backlash_nonlinearity(b)     # backlash nonlinearity with width b
relay_hysteresis_nonlinearity(b, c)   # relay output of amplitude b with
                                      # hysteresis of half-width c
saturation_nonlinearity(ub[, lb])     # saturation nonlinearity with upper
                                      # bound and (optional) lower bound

Calling these functions will create an object F that can be used for describing function analysis. For example, to create a saturation nonlinearity:

F = ct.saturation_nonlinearity(1)

These functions use the :class:`~control.DescribingFunctionNonlinearity`, which allows an analytical description of the describing function.

.. autosummary::
   :toctree: generated/
   :template: custom-class-template.rst

   ~control.DescribingFunctionNonlinearity
   ~control.friction_backlash_nonlinearity
   ~control.relay_hysteresis_nonlinearity
   ~control.saturation_nonlinearity