control.TransferFunction

class control.TransferFunction(num, den[, dt])

Bases: LTI

Transfer function representation for LTI input/output systems.

The TransferFunction class is used to represent systems in transfer function form. Transfer functions are usually created with the tf factory function.

Parameters

num2D list of coefficient arrays

Polynomial coefficients of the numerator.

den2D list of coefficient arrays

Polynomial coefficients of the denominator.

dtNone, True or float, optional

System timebase. 0 (default) indicates continuous time, True indicates discrete time with unspecified sampling time, positive number is discrete time with specified sampling time, None indicates unspecified timebase (either continuous or discrete time).

See also

tf, InputOutputSystem, FrequencyResponseData

Notes

The numerator and denominator polynomials are stored as 2D arrays with each element containing a 1D array of coefficients. These data structures can be retrieved using num_array and den_array. For example,

>>> sys.num_array[2, 5]         

gives the numerator of the transfer function from the 6th input to the 3rd output. (Note: a single 3D array structure cannot be used because the numerators and denominators can have different numbers of coefficients in each entry.)

The attributes num_list and den_list are properties that return 2D nested lists containing MIMO numerator and denominator coefficients. For example,

>>> sys.num_list[2][5]          

For legacy purposes, this list-based representation can also be obtained using num and den.

A discrete-time transfer function is created by specifying a nonzero ‘timebase’ dt when the system is constructed:

• dt = 0: continuous-time system (default)

• dt > 0: discrete-time system with sampling period dt

• dt = True: discrete time with unspecified sampling period

• dt = None: no timebase specified

Systems must have compatible timebases in order to be combined. A discrete-time system with unspecified sampling time (dt = True) can be combined with a system having a specified sampling time; the result will be a discrete-time system with the sample time of the other system. Similarly, a system with timebase None can be combined with a system having any timebase; the result will have the timebase of the other system. The default value of dt can be changed by changing the value of config.defaults['control.default_dt'].

A transfer function is callable and returns the value of the transfer function evaluated at a point in the complex plane. See TransferFunction.__call__ for a more detailed description.

Subsystems corresponding to selected input/output pairs can be created by indexing the transfer function:

subsys = sys[output_spec, input_spec]

The input and output specifications can be single integers, lists of integers, or slices. In addition, the strings representing the names of the signals can be used and will be replaced with the equivalent signal offsets.

The TransferFunction class defines two constants s and z that represent the differentiation and delay operators in continuous and discrete time. These can be used to create variables that allow algebraic creation of transfer functions. For example,

>>> s = ct.TransferFunction.s  # or ct.tf('s')
>>> G = (s + 1)/(s**2 + 2*s + 1)

Attributes

ninputs, noutputsint

Number of input and output signals.

shapetuple

2-tuple of I/O system dimension, (noutputs, ninputs).

input_labels, output_labelslist of str

Names for the input and output signals.

namestring, optional

System name.

num_array, den_array2D array of lists of float

Numerator and denominator polynomial coefficients as 2D array of 1D array objects (of varying length).

num_list, den_list2D list of 1D array

Numerator and denominator polynomial coefficients as 2D lists of 1D array objects (of varying length).

display_formatNone, ‘poly’ or ‘zpk’

Display format used in printing the TransferFunction object. Default behavior is polynomial display and can be changed by changing config.defaults['xferfcn.display_format'].

sTransferFunction

Differentiation operator (continuous time).

zTransferFunction

Delay operator (discrete time).

Attributes

den	Denominator polynomial (as 2D nested lists of 1D arrays).
den_array	Denominator polynomial coefficients as a 2D array of 1D coefficients.
den_list	Denominator polynomial (as 2D nested lists of 1D arrays).
dt	System timebase.
input_labels	List of labels for the input signals.
ninputs	Number of system inputs.
noutputs	Number of system outputs.
nstates	Number of system states.
num	Numerator polynomial (as 2D nested list of 1D arrays).
num_array	Numerator polynomial coefficients as a 2D array of 1D coefficients.
num_list	Numerator polynomial (as 2D nested list of 1D arrays).
output_labels	List of labels for the output signals.
repr_format	String representation format.
s	Differentiation operator (continuous time).
shape	2-tuple of I/O system dimension, (noutputs, ninputs).
state_labels	List of labels for the state signals.
z	Delay operator (discrete time).

Methods

__call__	Evaluate system transfer function at point in complex plane.
append	Append a second model to the present model.
bandwidth	Evaluate bandwidth of an LTI system for a given dB drop.
bode_plot	Generate a Bode plot for the system.
copy	Make a copy of an input/output system.
damp	Natural frequency, damping ratio of system poles.
dcgain	Return the zero-frequency ("DC") gain.
feedback	Feedback interconnection between two LTI objects.
find_input	Find the index for an input given its name (None if not found).
find_inputs	Return list of indices matching input spec (None if not found).
find_output	Find the index for a output given its name (None if not found).
find_outputs	Return list of indices matching output spec (None if not found).
find_state	Find the index for a state given its name (None if not found).
find_states	Return list of indices matching state spec (None if not found).
forced_response	Generate the forced response for the system.
freqresp	Evaluate transfer function at complex frequencies.
frequency_response	Evaluate LTI system response at an array of frequencies.
horner	Evaluate value of transfer function using Horner's method.
impulse_response	Generate the impulse response for the system.
initial_response	Generate the initial response for the system.
isctime	Check to see if a system is a continuous-time system.
isdtime	Check to see if a system is a discrete-time system.
ispassive	Indicate if a linear time invariant (LTI) system is passive.
issiso	Check to see if a system is single input, single output.
minreal	Remove canceling pole/zero pairs from a transfer function.
nichols_plot	Generate a Nichols plot for the system.
nyquist_plot	Generate a Nyquist plot for the system.
poles	Compute the poles of a transfer function.
returnScipySignalLTI	Return a 2D array of scipy.signal.lti objects.
sample	Convert a continuous-time system to discrete time.
set_inputs	Set the number/names of the system inputs.
set_outputs	Set the number/names of the system outputs.
set_states	Set the number/names of the system states.
step_response	Generate the step response for the system.
to_ss	Convert to state space representation.
to_tf	Convert to transfer function representation.
update_names	Update signal and system names for an I/O system.
zeros	Compute the zeros of a transfer function.

__add__(other)

Add two LTI objects (parallel connection).

__call__(x, squeeze=None, warn_infinite=True)

Evaluate system transfer function at point in complex plane.

Returns the value of the system’s transfer function at a point x in the complex plane, where x is s for continuous-time systems and z for discrete-time systems.

See LTI.__call__ for details.

__mul__(other)

Multiply two LTI objects (serial connection).

__neg__()

Negate a transfer function.

__radd__(other)

Right add two LTI objects (parallel connection).

__rmul__(other)

Right multiply two LTI objects (serial connection).

__rsub__(other)

Right subtract two LTI objects.

__rtruediv__(other)

Right divide two LTI objects.

__sub__(other)

Subtract two LTI objects.

__truediv__(other)

Divide two LTI objects.

append(other)

Append a second model to the present model.

The second model is converted to a transfer function if necessary, inputs and outputs are appended and their order is preserved.

Parameters

otherStateSpace or TransferFunction

System to be appended.

Returns

sysTransferFunction

System model with other appended to self.

bandwidth(dbdrop=-3)

Evaluate bandwidth of an LTI system for a given dB drop.

Evaluate the first frequency that the response magnitude is lower than DC gain by dbdrop dB.

Parameters

dbdropfloat, optional

A strictly negative scalar in dB (default = -3) defines the amount of gain drop for deciding bandwidth.

Returns

bandwidthndarray

The first frequency (rad/time-unit) where the gain drops below dbdrop of the dc gain of the system, or nan if the system has infinite dc gain, inf if the gain does not drop for all frequency.

Raises

TypeError

If sys is not an SISO LTI instance.

ValueError

If dbdrop is not a negative scalar.

bode_plot(*args, **kwargs)

Generate a Bode plot for the system.

See bode_plot for more information.

copy(name=None, use_prefix_suffix=True)

Make a copy of an input/output system.

A copy of the system is made, with a new name. The name keyword can be used to specify a specific name for the system. If no name is given and use_prefix_suffix is True, the name is constructed by prepending config.defaults['iosys.duplicate_system_name_prefix'] and appending config.defaults['iosys.duplicate_system_name_suffix']. Otherwise, a generic system name of the form ‘sys[<id>]’ is used, where ‘<id>’ is based on an internal counter.

Parameters

namestr, optional

Name of the newly created system.

use_prefix_suffixbool, optional

If True and name is None, set the name of the new system to the name of the original system with prefix config.defaults['duplicate_system_name_prefix'] and suffix config.defaults['duplicate_system_name_suffix'].

Returns

InputOutputSystem

damp()

Natural frequency, damping ratio of system poles.

Returns

wnarray

Natural frequency for each system pole.

zetaarray

Damping ratio for each system pole.

polesarray

System pole locations.

dcgain(warn_infinite=False)

Return the zero-frequency (“DC”) gain.

For a continuous-time transfer function G(s), the DC gain is G(0) For a discrete-time transfer function G(z), the DC gain is G(1)

Parameters

warn_infinitebool, optional

By default, don’t issue a warning message if the zero-frequency gain is infinite. Setting warn_infinite to generate the warning message.

Returns

gain(noutputs, ninputs) ndarray or scalar

Array or scalar value for SISO systems, depending on config.defaults['control.squeeze_frequency_response']. The value of the array elements or the scalar is either the zero-frequency (or DC) gain, or inf, if the frequency response is singular.

For real valued systems, the empty imaginary part of the complex zero-frequency response is discarded and a real array or scalar is returned.

Examples

>>> G = ct.tf([1], [1, 4])
>>> G.dcgain()
np.float64(0.25)

property den

Denominator polynomial (as 2D nested lists of 1D arrays).

den_array

Denominator polynomial coefficients as a 2D array of 1D coefficients.

property den_list

Denominator polynomial (as 2D nested lists of 1D arrays).

dt

System timebase.

feedback(other=1, sign=-1)

Feedback interconnection between two LTI objects.

Parameters

otherInputOutputSystem

System in the feedback path.

signfloat, optional

Gain to use in feedback path. Defaults to -1.

find_input(name)

Find the index for an input given its name (None if not found).

Parameters

namestr

Signal name for the desired input.

Returns

int

Index of the named input.

find_inputs(name_list)

Return list of indices matching input spec (None if not found).

Parameters

name_liststr or list of str

List of signal specifications for the desired inputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int

List of indices for the specified inputs.

find_output(name)

Find the index for a output given its name (None if not found).

Parameters

namestr

Signal name for the desired output.

Returns

int

Index of the named output.

find_outputs(name_list)

Return list of indices matching output spec (None if not found).

Parameters

name_liststr or list of str

List of signal specifications for the desired outputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int

List of indices for the specified outputs.

find_state(name)

Find the index for a state given its name (None if not found).

Parameters

namestr

Signal name for the desired state.

Returns

int

Index of the named state.

find_states(name_list)

Return list of indices matching state spec (None if not found).

Parameters

name_liststr or list of str

List of signal specifications for the desired states. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int

List of indices for the specified states..

forced_response(*args, **kwargs)

Generate the forced response for the system.

See forced_response for more information.

freqresp(omega)

Evaluate transfer function at complex frequencies.

frequency_response(omega=None, squeeze=None)

Evaluate LTI system response at an array of frequencies.

See frequency_response for more detailed information.

horner(x, warn_infinite=True)

Evaluate value of transfer function using Horner’s method.

Evaluates sys(x) where x is a complex number s for continuous-time systems and z for discrete-time systems. Expects inputs and outputs to be formatted correctly. Use sys(x) for a more user-friendly interface.

Parameters

xcomplex

Complex frequency at which the transfer function is evaluated.

warn_infinitebool, optional

If True (default), generate a warning if x is a pole.

Returns

complex

impulse_response(*args, **kwargs)

Generate the impulse response for the system.

See impulse_response for more information.

initial_response(*args, **kwargs)

Generate the initial response for the system.

See initial_response for more information.

property input_labels

List of labels for the input signals.

isctime(strict=False)

Check to see if a system is a continuous-time system.

Parameters

strictbool, optional

If strict is True, make sure that timebase is not None. Default is False.

isdtime(strict=False)

Check to see if a system is a discrete-time system.

Parameters

strictbool, optional

If strict is True, make sure that timebase is not None. Default is False.

ispassive()

Indicate if a linear time invariant (LTI) system is passive.

See ispassive for details.

issiso()

Check to see if a system is single input, single output.

minreal(tol=None)

Remove canceling pole/zero pairs from a transfer function.

Parameters

tolfloat

Tolerance for determining whether poles and zeros overlap.

nichols_plot(*args, **kwargs)

Generate a Nichols plot for the system.

See nichols_plot for more information.

ninputs

Number of system inputs.

noutputs

Number of system outputs.

nstates

Number of system states.

property num

Numerator polynomial (as 2D nested list of 1D arrays).

num_array

Numerator polynomial coefficients as a 2D array of 1D coefficients.

property num_list

Numerator polynomial (as 2D nested list of 1D arrays).

nyquist_plot(*args, **kwargs)

Generate a Nyquist plot for the system.

See nyquist_plot for more information.

property output_labels

List of labels for the output signals.

poles()

Compute the poles of a transfer function.

property repr_format

String representation format.

Format used in creating the representation for the system:

• ‘info’ : <IOSystemType sysname: [inputs] -> [outputs]>

• ‘eval’ : system specific, loadable representation

• ‘latex’ : HTML/LaTeX representation of the object

The default representation for an input/output is set to ‘eval’. This value can be changed for an individual system by setting the repr_format parameter when the system is created or by setting the repr_format property after system creation. Set config.defaults['iosys.repr_format'] to change for all I/O systems or use the repr_format parameter/attribute for a single system.

returnScipySignalLTI(strict=True)

Return a 2D array of scipy.signal.lti objects.

For instance,

>>> out = tfobject.returnScipySignalLTI()               
>>> out[3, 5]                                           

is a scipy.signal.lti object corresponding to the transfer function from the 6th input to the 4th output.

Parameters

strictbool, optional

True (default):

The timebase tfobject.dt cannot be None; it must be continuous (0) or discrete (True or > 0).

False:

if tfobject.dt is None, continuous-time scipy.signal.lti objects are returned

Returns

outlist of list of scipy.signal.TransferFunction

Continuous time (inheriting from scipy.signal.lti) or discrete time (inheriting from scipy.signal.dlti) SISO objects.

s

Differentiation operator (continuous time).

The s constant can be used to create continuous-time transfer functions using algebraic expressions.

Examples

>>> s = TransferFunction.s                               
>>> G  = (s + 1)/(s**2 + 2*s + 1)                        

sample(Ts, method='zoh', alpha=None, prewarp_frequency=None, name=None, copy_names=True, **kwargs)

Convert a continuous-time system to discrete time.

Creates a discrete-time system from a continuous-time system by sampling. Multiple methods of conversion are supported.

Parameters

Tsfloat

Sampling period.

method{‘gbt’, ‘bilinear’, ‘euler’, ‘backward_diff’, ‘zoh’, ‘matched’}

Method to use for sampling:

• ‘gbt’: generalized bilinear transformation

• ‘backward_diff’: Backwards difference (‘gbt’ with alpha=1.0)

• ‘bilinear’ (or ‘tustin’): Tustin’s approximation (‘gbt’ with alpha=0.5)

• ‘euler’: Euler (or forward difference) method (‘gbt’ with alpha=0)

• ‘matched’: pole-zero match method

• ‘zoh’: zero-order hold (default)

alphafloat within [0, 1]

The generalized bilinear transformation weighting parameter, which should only be specified with method = ‘gbt’, and is ignored otherwise. See scipy.signal.cont2discrete.

prewarp_frequencyfloat within [0, infinity)

The frequency [rad/s] at which to match with the input continuous- time system’s magnitude and phase (the gain=1 crossover frequency, for example). Should only be specified with method = ‘bilinear’ or ‘gbt’ with alpha = 0.5 and ignored otherwise.

namestring, optional

Set the name of the sampled system. If not specified and if copy_names is False, a generic name ‘sys[id]’ is generated with a unique integer id. If copy_names is True, the new system name is determined by adding the prefix and suffix strings in config.defaults['iosys.sampled_system_name_prefix'] and config.defaults['iosys.sampled_system_name_suffix'], with the default being to add the suffix ‘$sampled’.

copy_namesbool, Optional

If True, copy the names of the input signals, output signals, and states to the sampled system.

Returns

sysdTransferFunction system

Discrete-time system, with sample period Ts.

Other Parameters

inputsint, list of str or None, optional

Description of the system inputs. If not specified, the original system inputs are used. See InputOutputSystem for more information.

outputsint, list of str or None, optional

Description of the system outputs. Same format as inputs.

Notes

Available only for SISO systems. Uses scipy.signal.cont2discrete.

Examples

>>> sys = ct.tf(1, [1, 1])
>>> sysd = sys.sample(0.5, method='bilinear')

set_inputs(inputs, prefix='u')

Set the number/names of the system inputs.

Parameters

inputsint, list of str, or None

Description of the system inputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘u[i]’ (where the prefix ‘u’ can be changed using the optional prefix parameter).

prefixstring, optional

If inputs is an integer, create the names of the states using the given prefix (default = ‘u’). The names of the input will be of the form ‘prefix[i]’.

set_outputs(outputs, prefix='y')

Set the number/names of the system outputs.

Parameters

outputsint, list of str, or None

Description of the system outputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘y[i]’ (where the prefix ‘y’ can be changed using the optional prefix parameter).

prefixstring, optional

If outputs is an integer, create the names of the states using the given prefix (default = ‘y’). The names of the input will be of the form ‘prefix[i]’.

set_states(states, prefix='x')

Set the number/names of the system states.

Parameters

statesint, list of str, or None

Description of the system states. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘x[i]’ (where the prefix ‘x’ can be changed using the optional prefix parameter).

prefixstring, optional

If states is an integer, create the names of the states using the given prefix (default = ‘x’). The names of the input will be of the form ‘prefix[i]’.

property shape

2-tuple of I/O system dimension, (noutputs, ninputs).

property state_labels

List of labels for the state signals.

step_response(*args, **kwargs)

Generate the step response for the system.

See step_response for more information.

to_ss(*args, **kwargs)

Convert to state space representation.

See ss for details.

to_tf(*args, **kwargs)

Convert to transfer function representation.

See tf for details.

update_names([name, inputs, outputs, states])

Update signal and system names for an I/O system.

Parameters

namestr, optional

New system name.

inputslist of str, int, or None, optional

List of strings that name the individual input signals. If given as an integer or None, signal names default to the form ‘u[i]’. See InputOutputSystem for more information.

outputslist of str, int, or None, optional

Description of output signals; defaults to ‘y[i]’.

statesint, list of str, int, or None, optional

Description of system states; defaults to ‘x[i]’.

input_prefixstring, optional

Set the prefix for input signals. Default = ‘u’.

output_prefixstring, optional

Set the prefix for output signals. Default = ‘y’.

state_prefixstring, optional

Set the prefix for state signals. Default = ‘x’.

z

Delay operator (discrete time).

The z constant can be used to create discrete-time transfer functions using algebraic expressions.

Examples

>>> z = TransferFunction.z                               
>>> G  = 2 * z / (4 * z**3 + 3*z - 1)                    

zeros()

Compute the zeros of a transfer function.