pymor.models.transfer_function

Module Contents

class pymor.models.transfer_function.FactorizedTransferFunction(dim_input, dim_output, K, B, C, D, dK=None, dB=None, dC=None, dD=None, parameters={}, sampling_time=0, name=None)[source]

Bases: TransferFunction

Transfer functions in generalized coprime factor form.

This class describes input-output systems given by a transfer function of the form \(H(s, \mu) = \mathcal{C}(s, \mu) \mathcal{K}(s, \mu)^{-1} \mathcal{B}(s, \mu) + \mathcal{D}(s, \mu)\).

Parameters

dim_input

The number of inputs.

dim_output

The number of outputs.

K, B, C, D

Functions that take s and return an Operator.

dK, dB, dC, dD

Functions that take s and return an Operator that is the derivative of K, B, C, D (optional).

parameters

The Parameters of the transfer function.

sampling_time

0 if the system is continuous-time, otherwise a positive number that denotes the sampling time (in seconds).

name

Name of the system.

class pymor.models.transfer_function.TransferFunction(dim_input, dim_output, tf, dtf=None, parameters={}, sampling_time=0, name=None)[source]

Bases: pymor.core.cache.CacheableObject, pymor.parameters.base.ParametricObject

Class for systems represented by a transfer function.

This class describes input-output systems given by a (parametrized) transfer function \(H(s, \mu)\).

Parameters

dim_input

The number of inputs.

dim_output

The number of outputs.

tf

The transfer function H, given by a callable that takes a complex value s and, if parametric, a parameter value mu. The result of tf(s, mu) is a NumPy array of shape (dim_output, dim_input).

dtf

The complex derivative of H with respect to s (optional).

parameters

The Parameters of the transfer function.

sampling_time

0 if the system is continuous-time, otherwise a positive number that denotes the sampling time (in seconds).

name

Name of the system.

dim_input[source]

The number of inputs.

dim_output[source]

The number of outputs.

tf[source]

The transfer function.

dtf[source]

The complex derivative of the transfer function.

Methods

bode

Compute magnitudes and phases.

bode_plot

Draw the Bode plot for all input-output pairs.

eval_dtf

Evaluate the derivative of the transfer function.

eval_tf

Evaluate the transfer function.

freq_resp

Evaluate the transfer function on the imaginary axis.

h2_inner

Compute H2 inner product with an LTIModel.

h2_norm

Compute the H2-norm using quadrature.

mag_plot

Draw the magnitude plot.

cache_region = memory[source]
bode(self, w, mu=None)[source]

Compute magnitudes and phases.

Parameters

w

A sequence of angular frequencies at which to compute the transfer function.

mu

Parameter values for which to evaluate the transfer function.

Returns

mag

Transfer function magnitudes at frequencies in w, NumPy array of shape (len(w), self.dim_output, self.dim_input).

phase

Transfer function phases (in radians) at frequencies in w, NumPy array of shape (len(w), self.dim_output, self.dim_input).

bode_plot(self, w, mu=None, ax=None, Hz=False, dB=False, deg=True, **mpl_kwargs)[source]

Draw the Bode plot for all input-output pairs.

Parameters

w

A sequence of angular frequencies at which to compute the transfer function.

mu

Parameter values for which to evaluate the transfer function.

ax

Axis of shape (2 * self.dim_output, self.dim_input) to which to plot. If not given, matplotlib.pyplot.gcf is used to get the figure and create axis.

Hz

Should the frequency be in Hz on the plot.

dB

Should the magnitude be in dB on the plot.

deg

Should the phase be in degrees (otherwise in radians).

mpl_kwargs

Keyword arguments used in the matplotlib plot function.

Returns

artists

List of matplotlib artists added.

eval_dtf(self, s, mu=None)[source]

Evaluate the derivative of the transfer function.

Parameters

s

Laplace variable as a complex number.

mu

Parameter values.

Returns

Transfer function value as a 2D NumPy array.

eval_tf(self, s, mu=None)[source]

Evaluate the transfer function.

Parameters

s

Laplace variable as a complex number.

mu

Parameter values.

Returns

Transfer function value as a 2D NumPy array.

freq_resp(self, w, mu=None)[source]

Evaluate the transfer function on the imaginary axis.

Parameters

w

A sequence of angular frequencies at which to compute the transfer function.

mu

Parameter values for which to evaluate the transfer function.

Returns

tfw

Transfer function values at frequencies in w, NumPy array of shape (len(w), self.dim_output, self.dim_input).

h2_inner(self, lti)[source]

Compute H2 inner product with an LTIModel.

Uses the inner product formula based on the pole-residue form (see, e.g., Lemma 1 in [ABG10]). It assumes that self.tf is defined on -lti.poles().

Parameters

lti

LTIModel consisting of Operators that can be converted to NumPy arrays. The D operator is ignored.

Returns

inner

H2 inner product.

h2_norm(self, return_norm_only=True, **quad_kwargs)[source]

Compute the H2-norm using quadrature.

This method uses scipy.integrate.quad and makes no assumptions on the form of the transfer function. It only assumes that self.tf is defined over the imaginary axis.

By default, the absolute error tolerance in scipy.integrate.quad is set to zero (see its optional argument epsabs). It can be changed by using the epsabs keyword argument.

Parameters

return_norm_only

Whether to only return the approximate H2-norm.

quad_kwargs

Keyword arguments passed to scipy.integrate.quad.

Returns

norm

Computed H2-norm.

norm_relerr

Relative error estimate (returned if return_norm_only is False).

info

Quadrature info (returned if return_norm_only is False and full_output is True). See scipy.integrate.quad documentation for more details.

mag_plot(self, w, mu=None, ax=None, ord=None, Hz=False, dB=False, **mpl_kwargs)[source]

Draw the magnitude plot.

Parameters

w

A sequence of angular frequencies at which to compute the transfer function.

mu

Parameter values for which to evaluate the transfer function.

ax

Axis to which to plot. If not given, matplotlib.pyplot.gca is used.

ord

The order of the norm used to compute the magnitude (the default is the Frobenius norm).

Hz

Should the frequency be in Hz on the plot.

dB

Should the magnitude be in dB on the plot.

mpl_kwargs

Keyword arguments used in the matplotlib plot function.

Returns

out

List of matplotlib artists added.