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 anInputOutputSystem
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
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 zerofrequency 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 timeinvariant 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 continuoustime system
isdtime
([strict])Check to see if a system is a discretetime 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 statespace 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 zerofrequency gain
The zerofrequency gain of a continuoustime statespace system is given by:
and of a discretetime statespace system by:
 Parameters
warn_infinite (bool, optional) – By default, don’t issue a warning message if the zerofrequency 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 zerofrequency (or DC) gain, or inf, if the frequency response is singular.
For real valued systems, the empty imaginary part of the complex zerofrequency 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 discretetime, 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
 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 timeinvariant 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 singledimensional 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 singledimensional 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 continuoustime systems and z for discretetime systems.
Expects inputs and outputs to be formatted correctly. Use
sys(x)
for a more userfriendly 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 fallback to python code.

isctime
(strict=False)¶ Check to see if a system is a continuoustime 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 discretetime 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=1e06, 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 fromscipy.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 discretetime system from a continuoustime 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: zeroorder 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
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 userfriendly 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.
