control.TransferFunction¶
- class control.TransferFunction(num, den[, dt])[source]¶
Bases:
LTI
A class for representing transfer functions.
The TransferFunction class is used to represent systems in transfer function form.
- Parameters
num (array_like, or list of list of array_like) – Polynomial coefficients of the numerator
den (array_like, or list of list of array_like) – Polynomial coefficients of the denominator
dt (None, 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).
display_format (None, 'poly' or 'zpk') – Set the display format used in printing the TransferFunction object. Default behavior is polynomial display and can be changed by changing config.defaults[‘xferfcn.display_format’].
- ninputs, noutputs, nstates
Number of input, output and state variables.
- Type
int
- num, den
Polynomial coefficients of the numerator and denominator.
- Type
2D list of array
- dt¶
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).
- Type
None, True or float
Notes
The attribues ‘num’ and ‘den’ are 2-D lists of arrays containing MIMO numerator and denominator coefficients. For example,
>>> num[2][5] = numpy.array([1., 4., 8.])
means that the numerator of the transfer function from the 6th input to the 3rd output is set to s^2 + 4s + 8.
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 latter system. Similarly, a system with timebase None can be combined with a system having any timebase; the result will have the timebase of the latter system. The default value of dt can be changed by changing the value of
control.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
__call__()
for a more detailed description.The TransferFunction class defines two constants
s
andz
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 >>> G = (s + 1)/(s**2 + 2*s + 1)
Methods
Evaluate the bandwidth of the LTI system for a given dB drop.
Make a copy of an input/output system
Natural frequency, damping ratio of system poles
Return the zero-frequency (or DC) gain
Feedback interconnection between two LTI objects.
Find the index for an input given its name (None if not found)
Find the index for an output given its name (None if not found)
Find the index for a state given its name (None if not found)
(deprecated) Evaluate transfer function at complex frequencies.
Evaluate the linear time-invariant system at an array of angular frequencies.
Evaluate system's transfer function at complex frequency using Horner's method.
Check to see if a system is a continuous-time system
Check to see if a system is a discrete-time system
ispassive
Check to see if a system is single input, single output
Remove cancelling pole/zero pairs from a transfer function
pole
Compute the poles of a transfer function.
Return a list of a list of
scipy.signal.lti
objects.Convert a continuous-time system to discrete time
Set the number/names of the system inputs.
Set the number/names of the system outputs.
Set the number/names of the system states.
zero
Compute the zeros of a transfer function.
- __call__(x, squeeze=None, warn_infinite=True)[source]¶
Evaluate system’s transfer function at complex frequencies.
Returns the complex frequency response sys(x) where x is s for continuous-time systems and z for discrete-time systems.
In general the system may be multiple input, multiple output (MIMO), where m = self.ninputs number of inputs and p = self.noutputs number of outputs.
To evaluate at a frequency omega in radians per second, enter
x = omega * 1j
, for continuous-time systems, orx = exp(1j * omega * dt)
for discrete-time systems. Or useTransferFunction.frequency_response()
.- Parameters
x (complex or complex 1D array_like) – Complex frequencies
squeeze (bool, optional) – If squeeze=True, remove single-dimensional entries from the shape of the output even if the system is not SISO. If squeeze=False, keep all indices (output, input and, if omega is array_like, frequency) even if the system is SISO. The default value can be set using config.defaults[‘control.squeeze_frequency_response’]. If True and the system is single-input single-output (SISO), return a 1D array rather than a 3D array. Default value (True) set by config.defaults[‘control.squeeze_frequency_response’].
warn_infinite (bool, optional) – If set to False, turn off divide by zero warning.
- Returns
fresp – The frequency response of the system. If the system is SISO and squeeze is not True, the shape of the array matches the shape of omega. If the system is not SISO or squeeze is False, the first two dimensions of the array are indices for the output and input and the remaining dimensions match omega. If
squeeze
is True then single-dimensional axes are removed.- Return type
complex ndarray
- bandwidth(dbdrop=-3)[source]¶
Evaluate the bandwidth of the LTI system for a given dB drop.
Evaluate the first frequency that the response magnitude is lower than DC gain by dbdrop dB.
- Parameters
dpdrop (float, optional) – A strictly negative scalar in dB (default = -3) defines the amount of gain drop for deciding bandwidth.
- Returns
bandwidth – 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
- Return type
ndarray
- Raises
TypeError – if ‘sys’ is not an SISO LTI instance
ValueError – if ‘dbdrop’ is not a negative scalar
- copy(name=None, use_prefix_suffix=True)[source]¶
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[‘namedio.duplicate_system_name_prefix’] and appending config.defaults[‘namedio.duplicate_system_name_suffix’]. Otherwise, a generic system name of the form sys[<id>] is used, where <id> is based on an internal counter.
- damp()[source]¶
Natural frequency, damping ratio of system poles
- Returns
wn (array) – Natural frequency for each system pole
zeta (array) – Damping ratio for each system pole
poles (array) – System pole locations
- dcgain(warn_infinite=False)[source]¶
Return the zero-frequency (or DC) gain
For a continous-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_infinite (bool, 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 – 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.
- Return type
(noutputs, ninputs) ndarray or scalar
Examples
>>> G = ct.tf([1], [1, 4]) >>> G.dcgain() 0.25
- den¶
Transfer function denominator polynomial (array)
The numerator of the transfer function is store as an 2D list of arrays containing MIMO numerator coefficients, indexed by outputs and inputs. For example,
den[2][5]
is the array of coefficients for the denominator of the transfer function from the sixth input to the third output.
- frequency_response(omega, squeeze=None)[source]¶
Evaluate the linear time-invariant system at an array of angular frequencies.
Reports the frequency response of the system,
G(j*omega) = mag * exp(j*phase)
for continuous time systems. For discrete time systems, the response is evaluated around the unit circle such that
G(exp(j*omega*dt)) = mag * exp(j*phase).
In general the system may be multiple input, multiple output (MIMO), where m = self.ninputs number of inputs and p = self.noutputs number of outputs.
- Parameters
omega (float or 1D array_like) – A list, tuple, array, or scalar value of frequencies in radians/sec at which the system will be evaluated.
squeeze (bool, optional) – If squeeze=True, remove single-dimensional entries from the shape of the output even if the system is not SISO. If squeeze=False, keep all indices (output, input and, if omega is array_like, frequency) even if the system is SISO. The default value can be set using config.defaults[‘control.squeeze_frequency_response’].
- Returns
response – Frequency response data object representing the frequency response. This object can be assigned to a tuple using
mag, phase, omega = response
where
mag
is the magnitude (absolute value, not dB or log10) of the system frequency response,phase
is the wrapped phase in radians of the system frequency response, andomega
is the (sorted) frequencies at which the response was evaluated. If the system is SISO and squeeze is not True,magnitude
andphase
are 1D, indexed by frequency. If the system is not SISO or squeeze is False, the array is 3D, indexed by the output, input, and frequency. Ifsqueeze
is True then single-dimensional axes are removed.- Return type
FrequencyReponseData
- horner(x, warn_infinite=True)[source]¶
Evaluate system’s transfer function at complex frequency using Horner’s method.
Evaluates sys(x) where x is 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
x (complex array_like or complex scalar) – Complex frequencies
- Returns
output – Frequency response
- Return type
(self.noutputs, self.ninputs, len(x)) complex ndarray
- property inputs¶
Deprecated attribute; use
ninputs
instead.The
inputs
attribute was used to store the number of system inputs. It is no longer used. If you need access to the number of inputs for an LTI system, useninputs
.
- isctime(strict=False)[source]¶
Check to see if a system is a continuous-time system
- Parameters
sys (Named I/O system) – System to be checked
strict (bool, optional) – If strict is True, make sure that timebase is not None. Default is False.
- isdtime(strict=False)[source]¶
Check to see if a system is a discrete-time system
- Parameters
strict (bool, optional) – If strict is True, make sure that timebase is not None. Default is False.
- ninputs¶
Number of system inputs.
- noutputs¶
Number of system outputs.
- nstates¶
Number of system states.
- num¶
Transfer function numerator polynomial (array)
The numerator of the transfer function is stored as an 2D list of arrays containing MIMO numerator coefficients, indexed by outputs and inputs. For example,
num[2][5]
is the array of coefficients for the numerator of the transfer function from the sixth input to the third output.
- property outputs¶
Deprecated attribute; use
noutputs
instead.The
outputs
attribute was used to store the number of system outputs. It is no longer used. If you need access to the number of outputs for an LTI system, usenoutputs
.
- returnScipySignalLTI(strict=True)[source]¶
Return a list of a list 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
strict (bool, 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
out – continuous time (inheriting from
scipy.signal.lti
) or discrete time (inheriting fromscipy.signal.dlti
) SISO objects- Return type
list of list of
scipy.signal.TransferFunction
- s(num, den[, dt])¶
Differentation operator (continuous time)
The
s
constant can be used to create continuous time transfer functions using algebraic expressions.Example
>>> 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)[source]¶
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
Ts (float) – Sampling period
method ({"gbt", "bilinear", "euler", "backward_diff",) –
“zoh”, “matched”} Method to use for sampling:
gbt: generalized bilinear transformation
bilinear or tustin: Tustin’s approximation (“gbt” with alpha=0.5)
euler: Euler (or forward difference) method (“gbt” with alpha=0)
backward_diff: Backwards difference (“gbt” with alpha=1.0)
zoh: zero-order hold (default)
alpha (float 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_frequency (float 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.
name (string, 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[‘namedio.sampled_system_name_prefix’] and config.defaults[‘namedio.sampled_system_name_suffix’], with the default being to add the suffix ‘$sampled’.
copy_names (bool, Optional) – If True, copy the names of the input signals, output signals, and states to the sampled system.
inputs (int, list of str or None, optional) – Description of the system inputs. If not specified, the origional system inputs are used. See
InputOutputSystem
for more information.outputs (int, list of str or None, optional) – Description of the system outputs. Same format as inputs.
- Returns
sysd – Discrete-time system, with sample period Ts
- Return type
TransferFunction system
Notes
Available only for SISO systems
Examples
>>> sys = ct.tf(1, [1, 1]) >>> sysd = sys.sample(0.5, method='bilinear')
- set_inputs(inputs, prefix='u')[source]¶
Set the number/names of the system inputs.
- Parameters
inputs (int, 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).
prefix (string, 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')[source]¶
Set the number/names of the system outputs.
- Parameters
outputs (int, 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 u[i] (where the prefix u can be changed using the optional prefix parameter).
prefix (string, 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')[source]¶
Set the number/names of the system states.
- Parameters
states (int, 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 u[i] (where the prefix u can be changed using the optional prefix parameter).
prefix (string, 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].
- z(num, den[, dt])¶
Delay operator (discrete time)
The
z
constant can be used to create discrete time transfer functions using algebraic expressions.Example
>>> z = TransferFunction.z >>> G = 2 * z / (4 * z**3 + 3*z - 1)