control.LTI

class control.LTI(inputs=1, outputs=1, states=None, name=None, **kwargs)

Bases: InputOutputSystem

Parent class for linear time-invariant system objects.

LTI is the parent to the FrequencyResponseData, StateSpace, and TransferFunction child classes. It contains the number of inputs and outputs, and the timebase (dt) for the system. This class is not generally accessed directly by the user.

See also

InputOutputSystem, StateSpace, TransferFunction, FrequencyResponseData

Attributes

dt	System timebase.
input_labels	List of labels for the input signals.
ninputs	Number of system inputs.
noutputs	Number of system outputs.
nstates	Number of system states.
output_labels	List of labels for the output signals.
repr_format	String representation format.
shape	2-tuple of I/O system dimension, (noutputs, ninputs).
state_labels	List of labels for the state signals.

Methods

__call__	Evaluate system transfer function at point in complex plane.
bandwidth	Evaluate bandwidth of an LTI system for a given dB drop.
bode_plot	Generate a Bode plot for the system.
copy	Make a copy of an input/output system.
damp	Natural frequency, damping ratio of system poles.
dcgain	Return the zero-frequency (DC) gain.
feedback	Feedback interconnection between two input/output systems.
find_input	Find the index for an input given its name (None if not found).
find_inputs	Return list of indices matching input spec (None if not found).
find_output	Find the index for a output given its name (None if not found).
find_outputs	Return list of indices matching output spec (None if not found).
find_state	Find the index for a state given its name (None if not found).
find_states	Return list of indices matching state spec (None if not found).
forced_response	Generate the forced response for the system.
frequency_response	Evaluate LTI system response at an array of frequencies.
impulse_response	Generate the impulse response for the system.
initial_response	Generate the initial response for the system.
isctime	Check to see if a system is a continuous-time system.
isdtime	Check to see if a system is a discrete-time system.
ispassive	Indicate if a linear time invariant (LTI) system is passive.
issiso	Check to see if a system is single input, single output.
nichols_plot	Generate a Nichols plot for the system.
nyquist_plot	Generate a Nyquist plot for the system.
set_inputs	Set the number/names of the system inputs.
set_outputs	Set the number/names of the system outputs.
set_states	Set the number/names of the system states.
step_response	Generate the step response for the system.
to_ss	Convert to state space representation.
to_tf	Convert to transfer function representation.
update_names	Update signal and system names for an I/O system.

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

Evaluate system transfer function at point in complex plane.

Returns the value of the system’s transfer function at a point x in the complex plane, where x is s for continuous-time systems and z for discrete-time systems.

By default, a (complex) scalar will be returned for SISO systems and a p x m array will be return for MIMO systems with m inputs and p outputs. This can be changed using the squeeze keyword.

To evaluate at a frequency omega in radians per second, enter x = omega * 1j for continuous-time systems, x = exp(1j * omega * dt) for discrete-time systems, or use the frequency_response method.

Parameters:

xcomplex or complex 1D array_like

Complex value(s) at which transfer function will be evaluated.

squeezebool, optional

Squeeze output, as described below. Default value can be set using config.defaults['control.squeeze_frequency_response'].

warn_infinitebool, optional

If set to False, turn off divide by zero warning.

Returns:

frespcomplex ndarray

The value of the system transfer function at x. If the system is SISO and squeeze is not True, the shape of the array matches the shape of x. 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 x. If squeeze is True then single-dimensional axes are removed.

Notes

See FrequencyResponseData.__call__, StateSpace.__call__, TransferFunction.__call__ for class-specific details.

bandwidth(dbdrop=-3)

Evaluate bandwidth of an LTI system for a given dB drop.

Evaluate the first frequency that the response magnitude is lower than DC gain by dbdrop dB.

Parameters:

dbdropfloat, optional

A strictly negative scalar in dB (default = -3) defines the amount of gain drop for deciding bandwidth.

Returns:

bandwidthndarray

The first frequency (rad/time-unit) where the gain drops below dbdrop of the dc gain of the system, or nan if the system has infinite dc gain, inf if the gain does not drop for all frequency.

Raises:

TypeError

If sys is not an SISO LTI instance.

ValueError

If dbdrop is not a negative scalar.

bode_plot(*args, **kwargs)

Generate a Bode plot for the system.

See bode_plot for more information.

copy(name=None, use_prefix_suffix=True)

Make a copy of an input/output system.

A copy of the system is made, with a new name. The name keyword can be used to specify a specific name for the system. If no name is given and use_prefix_suffix is True, the name is constructed by prepending config.defaults['iosys.duplicate_system_name_prefix'] and appending config.defaults['iosys.duplicate_system_name_suffix']. Otherwise, a generic system name of the form ‘sys[<id>]’ is used, where ‘<id>’ is based on an internal counter.

Parameters:

namestr, optional

Name of the newly created system.

use_prefix_suffixbool, optional

If True and name is None, set the name of the new system to the name of the original system with prefix config.defaults['duplicate_system_name_prefix'] and suffix config.defaults['duplicate_system_name_suffix'].

Returns:

InputOutputSystem

damp()

Natural frequency, damping ratio of system poles.

Returns:

wnarray

Natural frequency for each system pole.

zetaarray

Damping ratio for each system pole.

polesarray

System pole locations.

dcgain()

Return the zero-frequency (DC) gain.

dt

System timebase.

feedback(other=1, sign=-1)

Feedback interconnection between two input/output systems.

Parameters:

otherInputOutputSystem

System in the feedback path.

signfloat, optional

Gain to use in feedback path. Defaults to -1.

find_input(name)

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

Parameters:

namestr

Signal name for the desired input.

Returns:

int

Index of the named input.

find_inputs(name_list)

Return list of indices matching input spec (None if not found).

Parameters:

name_liststr or list of str

List of signal specifications for the desired inputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns:

list of int

List of indices for the specified inputs.

find_output(name)

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

Parameters:

namestr

Signal name for the desired output.

Returns:

int

Index of the named output.

find_outputs(name_list)

Return list of indices matching output spec (None if not found).

Parameters:

name_liststr or list of str

List of signal specifications for the desired outputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns:

list of int

List of indices for the specified outputs.

find_state(name)

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

Parameters:

namestr

Signal name for the desired state.

Returns:

int

Index of the named state.

find_states(name_list)

Return list of indices matching state spec (None if not found).

Parameters:

name_liststr or list of str

List of signal specifications for the desired states. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns:

list of int

List of indices for the specified states..

forced_response(*args, **kwargs)

Generate the forced response for the system.

See forced_response for more information.

frequency_response(omega=None, squeeze=None)

Evaluate LTI system response at an array of frequencies.

See frequency_response for more detailed information.

impulse_response(*args, **kwargs)

Generate the impulse response for the system.

See impulse_response for more information.

initial_response(*args, **kwargs)

Generate the initial response for the system.

See initial_response for more information.

property input_labels

List of labels for the input signals.

isctime(strict=False)

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

Parameters:

strictbool, 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:

strictbool, optional

If strict is True, make sure that timebase is not None. Default is False.

ispassive()

Indicate if a linear time invariant (LTI) system is passive.

See ispassive for details.

issiso()

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

nichols_plot(*args, **kwargs)

Generate a Nichols plot for the system.

See nichols_plot for more information.

ninputs

Number of system inputs.

noutputs

Number of system outputs.

nstates

Number of system states.

nyquist_plot(*args, **kwargs)

Generate a Nyquist plot for the system.

See nyquist_plot for more information.

property output_labels

List of labels for the output signals.

property repr_format

String representation format.

Format used in creating the representation for the system:

• ‘info’ : <IOSystemType sysname: [inputs] -> [outputs]>

• ‘eval’ : system specific, loadable representation

• ‘latex’ : HTML/LaTeX representation of the object

The default representation for an input/output is set to ‘eval’. This value can be changed for an individual system by setting the repr_format parameter when the system is created or by setting the repr_format property after system creation. Set config.defaults['iosys.repr_format'] to change for all I/O systems or use the repr_format parameter/attribute for a single system.

set_inputs(inputs, prefix='u')

Set the number/names of the system inputs.

Parameters:

inputsint, 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).

prefixstring, 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:

outputsint, 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 ‘y[i]’ (where the prefix ‘y’ can be changed using the optional prefix parameter).

prefixstring, 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:

statesint, 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 ‘x[i]’ (where the prefix ‘x’ can be changed using the optional prefix parameter).

prefixstring, 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]’.

property shape

2-tuple of I/O system dimension, (noutputs, ninputs).

property state_labels

List of labels for the state signals.

step_response(*args, **kwargs)

Generate the step response for the system.

See step_response for more information.

to_ss(*args, **kwargs)

Convert to state space representation.

See ss for details.

to_tf(*args, **kwargs)

Convert to transfer function representation.

See tf for details.

update_names([name, inputs, outputs, states])

Update signal and system names for an I/O system.

Parameters:

namestr, optional

New system name.

inputslist of str, int, or None, optional

List of strings that name the individual input signals. If given as an integer or None, signal names default to the form ‘u[i]’. See InputOutputSystem for more information.

outputslist of str, int, or None, optional

Description of output signals; defaults to ‘y[i]’.

statesint, list of str, int, or None, optional

Description of system states; defaults to ‘x[i]’.

input_prefixstring, optional

Set the prefix for input signals. Default = ‘u’.

output_prefixstring, optional

Set the prefix for output signals. Default = ‘y’.

state_prefixstring, optional

Set the prefix for state signals. Default = ‘x’.