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.
Discretetime transfer functions are implemented by using the ‘dt’ instance variable and setting it to something other than ‘None’. If ‘dt’ has a nonzero value, then it must match whenever two transfer functions are combined. If ‘dt’ is set to True, the system will be treated as a discrete time system with unspecified sampling time. The default value of ‘dt’ is None and can be changed by changing the value of
control.config.defaults['xferfcn.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)¶ 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)TransferFunction(num, den[, dt]) damp
()Natural frequency, damping ratio of system poles dcgain
()Return the zerofrequency (or DC) gain evalfr
(omega)Evaluate a transfer function at a single angular frequency. feedback
([other, sign])Feedback interconnection between two LTI objects. freqresp
(omega)Evaluate the transfer function at a list of angular frequencies. horner
(s)Evaluate the systems’s transfer function for a complex variable 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
()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
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
()¶ 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)
Returns: gain – The zerofrequency gain Return type: ndarray

evalfr
(omega)¶ Evaluate a transfer function at a single angular frequency.
self._evalfr(omega) returns the value of the transfer function matrix with input value s = i * omega.

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

freqresp
(omega)¶ Evaluate the transfer function at a list of angular frequencies.
Reports the frequency response of the system,
G(j*omega) = mag*exp(j*phase)for continuous time. For discrete time systems, the response is evaluated around the unit circle such that
G(exp(j*omega*dt)) = mag*exp(j*phase).Parameters: omega (array_like) – A list of frequencies in radians/sec at which the system should be evaluated. The list can be either a python list or a numpy array and will be sorted before evaluation. Returns:  mag ((self.outputs, self.inputs, len(omega)) ndarray) – The magnitude (absolute value, not dB or log10) of the system frequency response.
 phase ((self.outputs, self.inputs, len(omega)) ndarray) – The wrapped phase in radians of the system frequency response.
 omega (ndarray or list or tuple) – The list of sorted frequencies at which the response was evaluated.

horner
(s)¶ Evaluate the systems’s transfer function for a complex variable
Returns a matrix of values evaluated at complex variable s.

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
()¶ Return a list of a list of scipy.signal.lti objects.
For instance,
>>> out = tfobject.returnScipySignalLTI() >>> out[3][5]
is a signal.scipy.lti object corresponding to the transfer function from the 6th input to the 4th output.

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: StateSpace system
Notes
 Available only for SISO systems
 Uses the command cont2discrete from scipy.signal
Examples
>>> sys = TransferFunction(1, [1,1]) >>> sysd = sys.sample(0.5, method='bilinear')

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