control.flatsys.FlatSystem

class control.flatsys.FlatSystem(forward, reverse, updfcn=None, outfcn=None, **kwargs)[source]

Bases: NonlinearIOSystem

Base class for representing a differentially flat system.

The FlatSystem class is used as a base class to describe differentially flat systems for trajectory generation. The output of the system does not need to be the differentially flat output. Flat systems are usually created with the flatsys factory function.

Parameters

forwardcallable: A function to compute the flat flag given the states and input.
reversecallable: A function to compute the states and input given the flat flag.
dtNone, True or float, optional: System timebase.

See also

flatsys

Notes

The class must implement two functions:

zflag = flatsys.forward(x, u, params)

This function computes the flag (derivatives) of the flat output. The inputs to this function are the state x and inputs u (both 1D arrays). The output should be a 2D array with the first dimension equal to the number of system inputs and the second dimension of the length required to represent the full system dynamics (typically the number of states)

x, u = flatsys.reverse(zflag, params)

This function system state and inputs give the the flag (derivatives) of the flat output. The input to this function is an 2D array whose first dimension is equal to the number of system inputs and whose second dimension is of length required to represent the full system dynamics (typically the number of states). The output is the state x and inputs u (both 1D arrays).

A flat system is also an input/output system supporting simulation, composition, and linearization. In the current implementation, the update function must be given explicitly, but the output function defaults to the flat outputs. If the output method is given, it is used in place of the flat outputs.

Attributes

ninputs, noutputs, nstatesint: Number of input, output and state variables.
shapetuple: 2-tuple of I/O system dimension, (noutputs, ninputs).
input_labels, output_labels, state_labelslist of str: Names for the input, output, and state variables.
namestring, optional: System name.

Attributes

`dt`	System timebase.
`input_labels`	List of labels for the input signals.
`ninputs`	Number of system inputs.
`noutputs`	Number of system outputs.
`nstates`	Number of system states.
`output_labels`	List of labels for the output signals.
`repr_format`	String representation format.
`shape`	2-tuple of I/O system dimension, (noutputs, ninputs).
`state_labels`	List of labels for the state signals.

Methods

`__call__`	Evaluate a (static) nonlinearity at a given input value.
`copy`	Make a copy of an input/output system.
`dynamics`	Dynamics of a differential or difference equation.
`feedback`	Feedback interconnection between two I/O systems.
`find_input`	Find the index for an input given its name (None if not found).
`find_inputs`	Return list of indices matching input spec (None if not found).
`find_output`	Find the index for a output given its name (None if not found).
`find_outputs`	Return list of indices matching output spec (None if not found).
`find_state`	Find the index for a state given its name (None if not found).
`find_states`	Return list of indices matching state spec (None if not found).
`forward`	Compute the flat flag given the states and input.
`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.
`linearize`	Linearize an input/output system at a given state and input.
`output`	Compute the output of the system.
`reverse`	Compute the states and input given the flat flag.
`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.
`update_names`	Update signal and system names for an I/O system.

__add__(other)[source]: Add two input/output systems (parallel interconnection)

__call__(u, params=None, squeeze=None)[source]

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

paramsdict, optional: Parameter values for the system. Passed to the evaluation function for the system as default values, overriding internal defaults.
squeezebool, 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'].

__mul__(other)[source]: Multiply two input/output systems (series interconnection)

__neg__()[source]: Negate an input/output system (rescale)

__radd__(other)[source]: Parallel addition of input/output system to a compatible object.

__rmul__(other)[source]: Pre-multiply an input/output systems by a scalar/matrix

__rsub__(other)[source]: Parallel subtraction of I/O system to a compatible object.

__sub__(other)[source]: Subtract two input/output systems (parallel interconnection)

__truediv__(other)[source]: Division of input/output system (by scalar or array)

copy(name=None, use_prefix_suffix=True)[source]

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.

Parameters

namestr, optional: Name of the newly created system.
use_prefix_suffixbool, optional: If True and name is None, set the name of the new system to the name of the original system with prefix config.defaults['duplicate_system_name_prefix'] and suffix config.defaults['duplicate_system_name_suffix'].

Returns

InputOutputSystem

dt

System timebase.

dynamics(t, x, u, params=None)[source]

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 a continuous-time system, returns the time derivative:

dx/dt = updfcn(t, x, u[, params])

where updfcn is the system’s (possibly nonlinear) update function. If the system is discrete time, returns the next value of x:

x[t+dt] = updfcn(t, x[t], u[t][, params])

where t is a scalar.

The inputs x and u must be of the correct length. The params argument is an optional dictionary of parameter values.

Parameters

tfloat: Time at which to evaluate.
xarray_like: Current state.
uarray_like: Current input.
paramsdict, optional: System parameter values.

Returns

dx/dt or x[t+dt]ndarray

feedback(other=1, sign=-1, params=None)[source]

Feedback interconnection between two I/O systems.

Parameters

otherInputOutputSystem: System in the feedback path.
signfloat, optional: Gain to use in feedback path. Defaults to -1.
paramsdict, optional: Parameter values for the overall system. Passed to the evaluation functions for the system as default values, overriding defaults for the individual systems.

Returns

NonlinearIOSystem

find_input(name)[source]

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

Parameters

namestr: Signal name for the desired input.

Returns

int: Index of the named input.

find_inputs(name_list)[source]

Return list of indices matching input spec (None if not found).

Parameters

name_liststr or list of str: List of signal specifications for the desired inputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int: List of indices for the specified inputs.

find_output(name)[source]

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

Parameters

namestr: Signal name for the desired output.

Returns

int: Index of the named output.

find_outputs(name_list)[source]

Return list of indices matching output spec (None if not found).

Parameters

name_liststr or list of str: List of signal specifications for the desired outputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int: List of indices for the specified outputs.

find_state(name)[source]

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

Parameters

namestr: Signal name for the desired state.

Returns

int: Index of the named state.

find_states(name_list)[source]

Return list of indices matching state spec (None if not found).

Parameters

name_liststr or list of str: List of signal specifications for the desired states. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.

Returns

list of int: List of indices for the specified states..

forward(x, u, params=None)[source]

Compute the flat flag given the states and input.

Given the states and inputs for a system, compute the flat outputs and their derivatives (the flat “flag”) for the system.

Parameters

xlist or array: The state of the system.
ulist or array: The input to the system.
paramsdict, optional: Parameter values for the system. Passed to the evaluation functions for the system as default values, overriding internal defaults.

Returns

zflaglist of 1D arrays: For each flat output $z_i$ , zflag[i] should be an ndarray of length $q_i$ that contains the flat output and its first $q_i$ derivatives.

property input_labels: List of labels for the input signals.

isctime(strict=False)[source]

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

Parameters

strictbool, optional: If strict is True, make sure that timebase is not None. Default is False.

isdtime(strict=False)[source]

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

Parameters

strictbool, optional: If strict is True, make sure that timebase is not None. Default is False.

issiso()[source]: Check to see if a system is single input, single output.

linearize(x0, u0=None, t=0, params=None, eps=1e-06, copy_names=False, **kwargs)[source]

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

Return the linearization of an input/output system at a given operating point (or state and input value) as a StateSpace system. See linearize for complete documentation.

ninputs

Number of system inputs.

noutputs

Number of system outputs.

nstates

Number of system states.

output(t, x, u, params=None)[source]

Compute the output of the system.

Given time t, input u and state x, returns the output of the system:

y = outfcn(t, x, u[, params])

The inputs x and u must be of the correct length.

Parameters

tfloat: The time at which to evaluate.
xarray_like: Current state.
uarray_like: Current input.
paramsdict, optional: System parameter values.

Returns

yndarray

property output_labels: List of labels for the output signals.

property repr_format

String representation format.

Format used in creating the representation for the system:

‘info’ : <IOSystemType sysname: [inputs] -> [outputs]>

‘eval’ : system specific, loadable representation

‘latex’ : HTML/LaTeX representation of the object

The default representation for an input/output is set to ‘eval’. This value can be changed for an individual system by setting the repr_format parameter when the system is created or by setting the repr_format property after system creation. Set config.defaults['iosys.repr_format'] to change for all I/O systems or use the repr_format parameter/attribute for a single system.

reverse(zflag, params=None)[source]

Compute the states and input given the flat flag.

Parameters

zflaglist of arrays: For each flat output $z_i$ , zflag[i] should be an ndarray of length $q_i$ that contains the flat output and its first $q_i$ derivatives.
paramsdict, optional: Parameter values for the system. Passed to the evaluation functions for the system as default values, overriding internal defaults.

Returns

x1D array: The state of the system corresponding to the flat flag.
u1D array: The input to the system corresponding to the flat flag.

set_inputs(inputs, prefix='u')[source]

Set the number/names of the system inputs.

Parameters

inputsint, 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).
prefixstring, 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')[source]

Set the number/names of the system outputs.

Parameters

outputsint, 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 ‘y[i]’ (where the prefix ‘y’ can be changed using the optional prefix parameter).
prefixstring, 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')[source]

Set the number/names of the system states.

Parameters

statesint, 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 ‘x[i]’ (where the prefix ‘x’ can be changed using the optional prefix parameter).
prefixstring, 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]’.

property shape: 2-tuple of I/O system dimension, (noutputs, ninputs).

property state_labels: List of labels for the state signals.

update_names([name, inputs, outputs, states])[source]

Update signal and system names for an I/O system.

Parameters

namestr, optional: New system name.
inputslist of str, int, or None, optional: List of strings that name the individual input signals. If given as an integer or None, signal names default to the form ‘u[i]’. See InputOutputSystem for more information.
outputslist of str, int, or None, optional: Description of output signals; defaults to ‘y[i]’.
statesint, list of str, int, or None, optional: Description of system states; defaults to ‘x[i]’.
input_prefixstring, optional: Set the prefix for input signals. Default = ‘u’.
output_prefixstring, optional: Set the prefix for output signals. Default = ‘y’.
state_prefixstring, optional: Set the prefix for state signals. Default = ‘x’.