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 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']`.

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)
```
`__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]) Natural frequency, damping ratio of system poles `dcgain`([warn_infinite]) Return the zero-frequency (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 time-invariant 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 continuous-time system `isdtime`([strict]) Check to see if a system is a discrete-time system Check to see if a system is single input, single output `minreal`([tol]) Remove cancelling pole/zero pairs from a transfer function 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 continuous-time system to discrete time 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 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

`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

`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

`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`

`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

Examples

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

Compute the zeros of a transfer function.