Describing functions

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.

Module usage

The function 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 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 describing_function_plot() function.

Pre-defined nonlinearities

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 DescribingFunctionNonlinearity, which allows an analytical description of the describing function.

Module classes and functions

`DescribingFunctionNonlinearity`()	Base class for nonlinear systems with a describing function.
`friction_backlash_nonlinearity`(b)	Backlash nonlinearity for describing function analysis.
`relay_hysteresis_nonlinearity`(b, c)	Relay w/ hysteresis nonlinearity for describing function analysis.
`saturation_nonlinearity`([ub, lb])	Create saturation nonlinearity for use in describing function analysis.