control.TransferFunction¶

class
control.
TransferFunction
(num, den[, dt])¶ A class for representing transfer functions
The TransferFunction class is used to represent systems in transfer function form.
The main data members are ‘num’ and ‘den’, which are 2D 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']
.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 = TransferFunction.s >>> G = (s + 1)/(s**2 + 2*s + 1)

__init__
(*args, **kwargs)¶ TransferFunction(num, den[, dt])
Construct a transfer function.
The default constructor is TransferFunction(num, den), where num and den are lists of lists of arrays containing polynomial coefficients. To create a discrete time transfer funtion, use TransferFunction(num, den, dt) where ‘dt’ is the sampling time (or True for unspecified sampling time). To call the copy constructor, call TransferFunction(sys), where sys is a TransferFunction object (continuous or discrete).
Methods
__init__
(*args, **kwargs)TransferFunction(num, den[, dt])
damp
()Natural frequency, damping ratio of system poles
dcgain
([warn_infinite])Return the zerofrequency (or DC) gain
feedback
([other, sign])Feedback interconnection between two LTI objects.
freqresp
(omega)(deprecated) Evaluate transfer function at complex frequencies.
frequency_response
(omega[, squeeze])Evaluate the linear timeinvariant system at an array of angular frequencies.
horner
(x[, warn_infinite])Evaluate system’s transfer function at complex frequency using Horner’s method.
isctime
([strict])Check to see if a system is a continuoustime system
isdtime
([strict])Check to see if a system is a discretetime system
issiso
()Check to see if a system is single input, single output
minreal
([tol])Remove cancelling pole/zero pairs from a transfer function
pole
()Compute the poles of a transfer function.
returnScipySignalLTI
([strict])Return a list of a list of
scipy.signal.lti
objects.sample
(Ts[, method, alpha, prewarp_frequency])Convert a continuoustime system to discrete time
zero
()Compute the zeros of a transfer function.
Attributes
inputs
outputs
s
z

damp
()¶ Natural frequency, damping ratio of system poles
 Returns
wn (array) – Natural frequencies for each system pole
zeta (array) – Damping ratio for each system pole
poles (array) – Array of system poles

dcgain
(warn_infinite=False)¶ Return the zerofrequency (or DC) gain
For a continoustime transfer function G(s), the DC gain is G(0) For a discretetime 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 zerofrequency 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 zerofrequency (or DC) gain, or inf, if the frequency response is singular.
For real valued systems, the empty imaginary part of the complex zerofrequency response is discarded and a real array or scalar is returned.
 Return type
(noutputs, ninputs) ndarray or scalar

feedback
(other=1, sign= 1)¶ Feedback interconnection between two LTI objects.

freqresp
(omega)¶ (deprecated) Evaluate transfer function at complex frequencies.

frequency_response
(omega, squeeze=None)¶ Evaluate the linear timeinvariant 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 singledimensional 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
mag (ndarray) – The magnitude (absolute value, not dB or log10) of the system frequency response. If the system is SISO and squeeze is not True, the array is 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. If
squeeze
is True then singledimensional axes are removed.phase (ndarray) – The wrapped phase in radians of the system frequency response.
omega (ndarray) – The (sorted) frequencies at which the response was evaluated.

horner
(x, warn_infinite=True)¶ Evaluate system’s transfer function at complex frequency using Horner’s method.
Evaluates sys(x) where x is s for continuoustime systems and z for discretetime systems.
Expects inputs and outputs to be formatted correctly. Use
sys(x)
for a more userfriendly 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

isctime
(strict=False)¶ Check to see if a system is a continuoustime system
 Parameters
sys (LTI 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)¶ Check to see if a system is a discretetime system
 Parameters
strict (bool, optional) – If strict is True, make sure that timebase is not None. Default is False.

issiso
()¶ Check to see if a system is single input, single output

minreal
(tol=None)¶ Remove cancelling pole/zero pairs from a transfer function

pole
()¶ Compute the poles of a transfer function.

returnScipySignalLTI
(strict=True)¶ 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

sample
(Ts, method='zoh', alpha=None, prewarp_frequency=None)¶ Convert a continuoustime system to discrete time
Creates a discretetime system from a continuoustime 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: 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: zeroorder 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.
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.
 Returns
sysd – Discrete time system, with sampling rate Ts
 Return type
TransferFunction system
Notes
Available only for SISO systems
Examples
>>> sys = TransferFunction(1, [1,1]) >>> sysd = sys.sample(0.5, method='bilinear')

zero
()¶ Compute the zeros of a transfer function.