control.describing_function_response

control.describing_function_response(H, F, A, omega=None, refine=True, warn_nyquist=None, _check_kwargs=True, **kwargs)[source]

Compute the describing function response of a system.

This function uses describing function analysis to analyze a closed loop system consisting of a linear system with a static nonlinear function in the feedback path.

Parameters

H (LTI system) – Linear time-invariant (LTI) system (state space, transfer function, or FRD)
F (static nonlinear function) – A static nonlinearity, either a scalar function or a single-input, single-output, static input/output system.
A (list) – List of amplitudes to be used for the describing function plot.
omega (list, optional) – List of frequencies to be used for the linear system Nyquist curve.
warn_nyquist (bool, optional) – Set to True to turn on warnings generated by nyquist_plot or False to turn off warnings. If not set (or set to None), warnings are turned off if omega is specified, otherwise they are turned on.
refine (bool, optional) – If True, scipy.optimize.minimize() to refine the estimate of the intersection of the frequency response and the describing function.

Returns

response (DescribingFunctionResponse object) – Response object that contains the result of the describing function analysis. The following information can be retrieved from this object:
response.intersections (1D array of 2-tuples or None) – A list of all amplitudes and frequencies in which $H(j\omega) N(a) = -1$ , where $N(a)$ is the describing function associated with F, or None if there are no such points. Each pair represents a potential limit cycle for the closed loop system with amplitude given by the first value of the tuple and frequency given by the second value.

Examples

>>> H_simple = ct.tf([8], [1, 2, 2, 1])
>>> F_saturation = ct.saturation_nonlinearity(1)
>>> amp = np.linspace(1, 4, 10)
>>> response = ct.describing_function_response(H_simple, F_saturation, amp)
>>> response.intersections  
[(3.343844998258643, 1.4142293090899216)]
>>> lines = response.plot()