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 as an output , where represents the (amplitude-dependent) gain and phase associated with the nonlinearity.
Stability analysis of a linear system with a feedback nonlinearity is done by looking for amplitudes and frequencies such that
If such an intersection exists, it indicates that there may be a limit cycle of amplitude with frequency .
Describing function analysis is a simple method, but it is approximate because it assumes that higher harmonics can be neglected.
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 and frequencies :math`omega` such that
These points can be determined by generating a Nyquist plot in which the
transfer function intersections the negative
reciprocal of the describing function . The
describing_function_plot() function generates this plot
and returns the amplitude and frequency of any points of intersection:
ct.describing_function_plot(H, F, amp_range[, omega_range])
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¶
Base class for nonlinear systems with a describing function
Backlash nonlinearity for use in describing function analysis
Relay w/ hysteresis nonlinearity for use in describing function analysis
Create a saturation nonlinearity for use in describing function analysis