# 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.

Discrete-time transfer functions are implemented by using the ‘dt’ instance variable and setting it to something other than ‘None’. If ‘dt’ has a non-zero 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` 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)

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]) Natural frequency, damping ratio of system poles Return the zero-frequency (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. Evaluate the systems’s transfer function for a complex variable returns True if and only if all of the numerator and denominator polynomials of the (possibly MIMO) transfer function are zeroth order, that is, if the system has no dynamics. `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. 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

 `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 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)

Returns

gain – The zero-frequency 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.

`is_static_gain`()

returns True if and only if all of the numerator and denominator polynomials of the (possibly MIMO) transfer function are zeroth order, that is, if the system has no dynamics.

`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`()

Return a list of a list of `scipy.signal.lti` objects.

For instance,

```>>> out = tfobject.returnScipySignalLTI()
>>> out[3][5]
```

is a class:scipy.signal.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 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

StateSpace 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.