control.LinearIOSystem

class control.LinearIOSystem(linsys, inputs=None, outputs=None, states=None, name=None, **kwargs)

Input/output representation of a linear (state space) system.

This class is used to implementat a system that is a linear state space system (defined by the StateSpace system object).

__init__(linsys, inputs=None, outputs=None, states=None, name=None, **kwargs)

Create an I/O system from a state space linear system.

Converts a StateSpace system into an InputOutputSystem with the same inputs, outputs, and states. The new system can be a continuous or discrete time system

Parameters

  • linsys (StateSpace) – LTI StateSpace system to be converted

  • inputs (int, list of str or None, optional) – Description of the system inputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form s[i] (where s is one of u, y, or x). If this parameter is not given or given as None, the relevant quantity will be determined when possible based on other information provided to functions using the system.

  • outputs (int, list of str or None, optional) – Description of the system outputs. Same format as inputs.

  • states (int, list of str, or None, optional) – Description of the system states. Same format as inputs.

  • 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).

  • params (dict, optional) – Parameter values for the systems. Passed to the evaluation functions for the system as default values, overriding internal defaults.

  • name (string, optional) – System name (used for specifying signals). If unspecified, a generic name <sys[id]> is generated with a unique integer id.

Returns

iosys – Linear system represented as an input/output system

Return type

LinearIOSystem

Methods

__init__(linsys[, inputs, outputs, states, name])

Create an I/O system from a state space linear system.

append(other)

Append a second model to the present model.

copy([newname])

Make a copy of an input/output system.

damp()

Natural frequency, damping ratio of system poles

dcgain([warn_infinite])

Return the zero-frequency gain

dynamics(t, x, u)

Compute the dynamics of a differential or difference equation.

feedback([other, sign, params])

Feedback interconnection between two input/output systems

find_input(name)

Find the index for an input given its name (None if not found)

find_output(name)

Find the index for an output given its name (None if not found)

find_state(name)

Find the index for a state given its name (None if not found)

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 Laub’s or 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

issiso()

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

lft(other[, nu, ny])

Return the Linear Fractional Transformation.

linearize(x0, u0[, t, params, eps, name, copy])

Linearize an input/output system at a given state and input.

minreal([tol])

Calculate a minimal realization, removes unobservable and uncontrollable states

name_or_default([name])

output(t, x, u)

Compute the output of the system

pole()

Compute the poles of a state space system.

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

set_inputs(inputs[, prefix])

Set the number/names of the system inputs.

set_outputs(outputs[, prefix])

Set the number/names of the system outputs.

set_states(states[, prefix])

Set the number/names of the system states.

slycot_laub(x)

Evaluate system’s transfer function at complex frequency using Laub’s method from Slycot.

zero()

Compute the zeros of a state space system.

Attributes

idCounter

inputs

outputs

states

append(other)

Append a second model to the present model.

The second model is converted to state-space if necessary, inputs and outputs are appended and their order is preserved

copy(newname=None)

Make a copy of an input/output system.

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 gain

The zero-frequency gain of a continuous-time state-space system is given by:

and of a discrete-time state-space system by:

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

dynamics(t, x, u)

Compute the dynamics of a differential or difference equation.

Given time t, input u and state x, returns the value of the right hand side of the dynamical system. If the system is continuous, returns the time derivative

dx/dt = f(t, x, u)

where f is the system’s (possibly nonlinear) dynamics function. If the system is discrete-time, returns the next value of x:

x[t+dt] = f(t, x[t], u[t])

Where t is a scalar.

The inputs x and u must be of the correct length.

Parameters

  • t (float) – the time at which to evaluate

  • x (array_like) – current state

  • u (array_like) – input

Returns

dx/dt or x[t+dt]

Return type

ndarray

feedback(other=1, sign=- 1, params={})

Feedback interconnection between two input/output systems

Parameters

  • sys1 (InputOutputSystem) – The primary process.

  • sys2 (InputOutputSystem) – The feedback process (often a feedback controller).

  • sign (scalar, optional) – The sign of feedback. sign = -1 indicates negative feedback, and sign = 1 indicates positive feedback. sign is an optional argument; it assumes a value of -1 if not specified.

Returns

out

Return type

InputOutputSystem

Raises

ValueError – if the inputs, outputs, or timebases of the systems are incompatible.

find_input(name)

Find the index for an input given its name (None if not found)

find_output(name)

Find the index for an output given its name (None if not found)

find_state(name)

Find the index for a state given its name (None if not found)

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 Laub’s or 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) – Complex frequencies

Returns

output – Frequency response

Return type

(self.noutputs, self.ninputs, len(x)) complex ndarray

Notes

Attempts to use Laub’s method from Slycot library, with a fall-back to python code.

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

lft(other, nu=- 1, ny=- 1)

Return the Linear Fractional Transformation.

A definition of the LFT operator can be found in Appendix A.7, page 512 in the 2nd Edition, Multivariable Feedback Control by Sigurd Skogestad.

An alternative definition can be found here: https://www.mathworks.com/help/control/ref/lft.html

Parameters

  • other (LTI) – The lower LTI system

  • ny (int, optional) – Dimension of (plant) measurement output.

  • nu (int, optional) – Dimension of (plant) control input.

linearize(x0, u0, t=0, params={}, eps=1e-06, name=None, copy=False, **kwargs)

Linearize an input/output system at a given state and input.

Return the linearization of an input/output system at a given state and input value as a StateSpace system. See linearize() for complete documentation.

minreal(tol=0.0)

Calculate a minimal realization, removes unobservable and uncontrollable states

output(t, x, u)

Compute the output of the system

Given time t, input u and state x, returns the output of the system:

y = g(t, x, u)

The inputs x and u must be of the correct length.

Parameters

  • t (float) – the time at which to evaluate

  • x (array_like) – current state

  • u (array_like) – input

Returns

y

Return type

ndarray

pole()

Compute the poles of a state space system.

returnScipySignalLTI(strict=True)

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

For instance,

>>> out = ssobject.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 ssobject.dt cannot be None; it must be continuous (0) or discrete (True or > 0).

False:

If ssobject.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.StateSpace

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"}) –

    Which method to use:

    • gbt: generalized bilinear transformation

    • bilinear: Tustin’s approximation (“gbt” with alpha=0.5)

    • euler: Euler (or forward differencing) method (“gbt” with alpha=0)

    • backward_diff: Backwards differencing (“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

Notes

Uses scipy.signal.cont2discrete()

Examples

>>> sys = StateSpace(0, 1, 1, 0)
>>> sysd = sys.sample(0.5, method='bilinear')
set_inputs(inputs, prefix='u')

Set the number/names of the system inputs.

Parameters

  • inputs (int, list of str, or None) – Description of the system inputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form u[i] (where the prefix u can be changed using the optional prefix parameter).

  • prefix (string, optional) – If inputs is an integer, create the names of the states using the given prefix (default = ‘u’). The names of the input will be of the form prefix[i].

set_outputs(outputs, prefix='y')

Set the number/names of the system outputs.

Parameters

  • outputs (int, list of str, or None) – Description of the system outputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form u[i] (where the prefix u can be changed using the optional prefix parameter).

  • prefix (string, optional) – If outputs is an integer, create the names of the states using the given prefix (default = ‘y’). The names of the input will be of the form prefix[i].

set_states(states, prefix='x')

Set the number/names of the system states.

Parameters

  • states (int, list of str, or None) – Description of the system states. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form u[i] (where the prefix u can be changed using the optional prefix parameter).

  • prefix (string, optional) – If states is an integer, create the names of the states using the given prefix (default = ‘x’). The names of the input will be of the form prefix[i].

slycot_laub(x)

Evaluate system’s transfer function at complex frequency using Laub’s method from Slycot.

Expects inputs and outputs to be formatted correctly. Use sys(x) for a more user-friendly interface.

Parameters

x (complex array_like or complex) – Complex frequency

Returns

output – Frequency response

Return type

(number_outputs, number_inputs, len(x)) complex ndarray

zero()

Compute the zeros of a state space system.