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 a second model to the present model.
Make a copy of an input/output system
Natural frequency, damping ratio of system poles
Return the zero-frequency gain
Compute the dynamics of a differential or difference equation.
Feedback interconnection between two input/output systems
Find the index for an input given its name (None if not found)
Find the index for an output given its name (None if not found)
Find the index for a state given its name (None if not found)
Compute the flat flag given the states and input.
(deprecated) Evaluate transfer function at complex frequencies.
Evaluate the linear time-invariant system at an array of angular frequencies.
Evaluate system's transfer function at complex frequency using Laub's or Horner's method.
Check to see if a system is a continuous-time system
Check to see if a system is a discrete-time system
Check to see if a system is single input, single output
Return the Linear Fractional Transformation.
Linearize an input/output system at a given state and input.
Calculate a minimal realization, removes unobservable and uncontrollable states
Compute the output of the system
pole
Compute the poles of a state space system.
Return a list of a list of
scipy.signal.lti
objects.Compute the states and input given the flat flag.
Convert a continuous time system to discrete time
Set the number/names of the system inputs.
Set the number/names of the system outputs.
Set the number/names of the system states.
Evaluate system's transfer function at complex frequency using Laub's method from Slycot.
zero
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
- 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, andomega
is the (sorted) frequencies at which the response was evaluated. If the system is SISO and squeeze is not True,magnitude
andphase
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. Ifsqueeze
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, useninputs
.
- 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, usenoutputs
.
- 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 fromscipy.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
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.