control.LinearICSystem

class control.LinearICSystem(io_sys, ss_sys=None)

Bases: control.iosys.InterconnectedSystem, control.iosys.LinearIOSystem

Interconnection of a set of linear input/output systems.

This class is used to implement a system that is an interconnection of linear input/output systems. It has all of the structure of an InterconnectedSystem, but also maintains the requirement elements of LinearIOSystem, including the StateSpace class structure, allowing it to be passed to functions that expect a StateSpace system.

This class is usually generated using interconnect() and not called directly

Methods

append

Append a second model to the present model.

copy

Make a copy of an input/output system.

damp

Natural frequency, damping ratio of system poles

dcgain

Return the zero-frequency gain

dynamics

Compute the dynamics of a differential or difference equation.

feedback

Feedback interconnection between two input/output systems

find_input

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

find_output

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

find_state

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

freqresp

(deprecated) Evaluate transfer function at complex frequencies.

frequency_response

Evaluate the linear time-invariant system at an array of angular frequencies.

horner

Evaluate system’s transfer function at complex frequency using Laub’s or Horner’s method.

isctime

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

isdtime

Check to see if a system is a discrete-time system

issiso

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

lft

Return the Linear Fractional Transformation.

linearize

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

minreal

Calculate a minimal realization, removes unobservable and uncontrollable states

output

Compute the output of the system

pole

Compute the poles of a state space system.

returnScipySignalLTI

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

sample

Convert a continuous time system to discrete time

set_connect_map

Set the connection map for an interconnected I/O system.

set_input_map

Set the input map for an interconnected I/O system.

set_inputs

Set the number/names of the system inputs.

set_output_map

Set the output map for an interconnected I/O system.

set_outputs

Set the number/names of the system outputs.

set_states

Set the number/names of the system states.

slycot_laub

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

zero

Compute the zeros of a state space system.

__add__(sys2)

Add two input/output systems (parallel interconnection)

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

Evaluate system’s transfer function at complex frequency.

Returns the complex frequency response sys(x) where x is s for continuous-time systems and z for discrete-time systems.

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

Parameters
  • x (complex or complex 1D array_like) – Complex frequencies

  • 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’].

  • warn_infinite (bool, optional) – If set to False, don’t warn if frequency response is infinite.

Returns

fresp – The frequency response of the system. If the system is SISO and squeeze is not True, the shape of the array matches the shape of omega. 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 omega. If squeeze is True then single-dimensional axes are removed.

Return type

complex ndarray

__div__(other)

Divide two LTI systems.

__getitem__(indices)

Array style access

__mul__(sys1)

Multiply two input/output systems (series interconnection)

__neg__()

Negate an input/output systems (rescale)

__radd__(other)

Right add two LTI systems (parallel connection).

__rdiv__(other)

Right divide two LTI systems.

__rmul__(sys2)

Pre-multiply an input/output systems by a scalar/matrix

__rsub__(other)

Right subtract two LTI systems.

__sub__(other)

Subtract two LTI systems.

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.

property inputs

Deprecated attribute; use ninputs instead.

The input attribute was used to store the number of system inputs. It is no longer used. If you need access to the number of inputs for an LTI system, use ninputs.

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

property outputs

Deprecated attribute; use noutputs instead.

The output attribute was used to store the number of system outputs. It is no longer used. If you need access to the number of outputs for an LTI system, use noutputs.

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_connect_map(connect_map)

Set the connection map for an interconnected I/O system.

Parameters

connect_map (2D array) – Specify the matrix that will be used to multiply the vector of subsystem outputs to obtain the vector of subsystem inputs.

set_input_map(input_map)

Set the input map for an interconnected I/O system.

Parameters

input_map (2D array) – Specify the matrix that will be used to multiply the vector of system inputs to obtain the vector of subsystem inputs. These values are added to the inputs specified in the connection map.

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_output_map(output_map)

Set the output map for an interconnected I/O system.

Parameters

output_map (2D array) – Specify the matrix that will be used to multiply the vector of subsystem outputs to obtain the vector of system outputs.

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

property states

Deprecated attribute; use nstates instead.

The state attribute was used to store the number of states for : a state space system. It is no longer used. If you need to access the number of states, use nstates.

zero()

Compute the zeros of a state space system.