control.flatsys.LinearFlatSystem¶

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

Bases: FlatSystem, LinearIOSystem

Base class for a linear, differentially flat system.

This class is used to create a differentially flat system representation from a linear 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. 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)

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)
`forward`	Compute the flat flag given the states and input.
`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`
`poles`	Compute the poles of a state space system.
`returnScipySignalLTI`	Return a list of a list of `scipy.signal.lti` objects.
`reverse`	Compute the states and input given the flat flag.
`sample`	Convert a continuous time system to discrete time
`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.
`slycot_laub`	Evaluate system's transfer function at complex frequency using Laub's method from Slycot.
`zero`
`zeros`	Compute the zeros of a state space system.

A¶

Dynamics matrix.

B¶

Input matrix.

C¶

Output matrix.

D¶

Direct term.

__add__(sys2)¶: Add two input/output systems (parallel interconnection)

__call__(u, params=None, squeeze=None)¶

Evaluate a (static) nonlinearity at a given input value

If a nonlinear I/O system has no internal state, then evaluating the system at an input u gives the output y = F(u), determined by the output function.

Parameters

params (dict, optional) – Parameter values for the system. Passed to the evaluation function for the system as default values, overriding internal defaults.
squeeze (bool, optional) – If True and if the system has a single output, return the system output as a 1D array rather than a 2D array. If False, return the system output as a 2D array even if the system is SISO. Default value set by config.defaults[‘control.squeeze_time_response’].

__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__(sys2)¶: Parallel addition of input/output system to a compatible object.

__rdiv__(other)¶: Right divide two LTI systems.

__rmul__(sys2)¶: Pre-multiply an input/output systems by a scalar/matrix

__rsub__(sys2)¶: Parallel subtraction of I/O system to a compatible object.

__sub__(sys2)¶: Subtract two input/output systems (parallel interconnection)

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(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.

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)

forward(x, u, params)¶

Compute the flat flag given the states and input.

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

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

response – Frequency response data object representing the frequency response. This object can be assigned to a tuple using

mag, phase, omega = response

where mag is the magnitude (absolute value, not dB or log10) of the system frequency response, phase is the wrapped phase in radians of the system frequency response, and omega is the (sorted) frequencies at which the response was evaluated. If the system is SISO and squeeze is not True, magnitude and phase are 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.

Return type

FrequencyReponseData

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 inputs 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 (Named I/O 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

ninputs¶

Number of system inputs.

noutputs¶

Number of system outputs.

nstates¶

Number of system 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 outputs 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.

poles()¶: 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

reverse(zflag, params)¶

Compute the states and input given the flat flag.

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

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

property states¶

zeros()¶: Compute the zeros of a state space system.