control.flatsys.LinearFlatSystem

class control.flatsys.LinearFlatSystem(linsys, inputs=None, outputs=None, states=None, name=None)
__init__(linsys, inputs=None, outputs=None, states=None, name=None)

Define a flat system from a SISO LTI system.

Given a reachable, single-input/single-output, linear time-invariant system, create a differentially flat system representation.

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. None (default) indicates continuous time, True indicates discrete time with undefined sampling time, positive number is discrete time with specified sampling 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)
Returns:

iosys – Linear system represented as an flat input/output system

Return type:

LinearFlatSystem

Methods

__init__(linsys[, inputs, outputs, states, name]) Define a flat system from a SISO LTI system.
append(other) 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
evalfr(omega) Evaluate a SS system’s transfer function at a single frequency.
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)
forward(x, u) Compute the flat flag given the states and input.
freqresp(omega) Evaluate the system’s transfer func.
horner(s) Evaluate the systems’s transfer function for a complex variable
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]) Linearize an input/output system at a given state and input.
minreal([tol]) Calculate a minimal realization, removes unobservable and uncontrollable states
pole() Compute the poles of a state space system.
returnScipySignalLTI() Return a list of a list of scipy.signal.lti objects.
reverse(zflag) Compute the states and input given the flat flag.
sample(Ts[, method, alpha]) 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.
zero() Compute the zeros of a state space system.
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()

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()

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:

Returns:gain – An array of shape (outputs,inputs); the array will either be the zero-frequency (or DC) gain, or, if the frequency response is singular, the array will be filled with np.nan.
Return type:ndarray
evalfr(omega)

Evaluate a SS system’s transfer function at a single frequency.

self._evalfr(omega) returns the value of the transfer function matrix with input value s = i * omega.

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)

forward(x, u)

Compute the flat flag given the states and input.

See control.flatsys.FlatSystem.forward() for more info.

freqresp(omega)

Evaluate the system’s transfer func. at a list of freqs, omega.

mag, phase, omega = self.freqresp(omega)

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) – 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 (float) – The magnitude (absolute value, not dB or log10) of the system frequency response.
  • phase (float) – The wrapped phase in radians of the system frequency response.
  • omega (array) – 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.

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)

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

pole()

Compute the poles of a state space system.

returnScipySignalLTI()

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

For instance,

>>> out = ssobject.returnScipySignalLTI()
>>> out[3][5]

is a signal.scipy.lti object corresponding to the transfer function from the 6th input to the 4th output.

reverse(zflag)

Compute the states and input given the flat flag.

See control.flatsys.FlatSystem.reverse() for more info.

sample(Ts, method='zoh', alpha=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
Returns:

sysd – Discrete time system, with sampling rate Ts

Return type:

StateSpace

Notes

Uses the command ‘cont2discrete’ from scipy.signal

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].
zero()

Compute the zeros of a state space system.