control.TransferFunction

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

Bases: control.lti.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).

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

Methods

damp

Natural frequency, damping ratio of system poles

dcgain

Return the zero-frequency (or DC) gain

feedback

Feedback interconnection between two LTI objects.

freqresp

(deprecated) Evaluate transfer function at complex frequencies.

frequency_response

Evaluate the linear time-invariant system at an array of angular frequencies.

horner

Evaluate system’s transfer function at complex frequency using Horner’s method.

isctime

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

isdtime

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

issiso

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

minreal

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

Convert a continuous-time system to discrete time

zero

Compute the zeros of a transfer function.

__add__(other)

Add two LTI objects (parallel connection).

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

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, or x = exp(1j * omega * dt) for discrete-time systems. Or use TransferFunction.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

__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.

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 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

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.

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 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

  • 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 single-dimensional 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 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 input 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, use ninputs.

isctime(strict=False)

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

ninputs

Number of system inputs.

noutputs

Number of system outputs.

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 output 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, use noutputs.

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 from scipy.signal.dlti) SISO objects

Return type

list of list of scipy.signal.TransferFunction

s

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)

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: 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.

  • 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

  1. Available only for SISO systems

  2. Uses scipy.signal.cont2discrete()

Examples

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

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)
zero()

Compute the zeros of a transfer function.