lmlib.statespace.model.AlssmExp#

class lmlib.statespace.model.AlssmExp(gamma, C=None, **kwargs)#

Bases: lmlib.statespace.model.ModelBase

ALSSM with discrete-time exponentially decaying/increasing output sequence.

Discrete-time linear state space model generating output sequences of exponentially decaying shape with decay factor \(\gamma\).

For more details see [Wildhaber2019] PDF

Parameters

gamma (float, int) – Decay factor per sample step ( > 1.0 : left-sided decaying; < 1.0 : right-sided decaying)
C (array_like, shape=(L, N), optional) – Output Matrix. If no output matrix is given, C gets initialize automatically to [[1]]. (default: C=None)
**kwargs – Forwarded to ModelBase

N : ALSSM system order, corresponding to the number of state variables
L : output order / number of signal channels

Examples

Parametrizing an exponentially ALSSM:

>>> gamma = 0.8
>>> alssm = lm.AlssmExp(gamma)
>>> print(alssm)
A =
[[0.8]]
C =
[[1.]]

Methods

`__init__`(gamma[, C])
`dump_tree`()	Returns the internal structure of the ALSSM model as a string.
`eval_state`(x)	Evaluation of the ALSSM for a state vector x.
`eval_states`(xs)	Evaluation of the ALSSM for an array of state vectors xs.
`get_state_var_indices`(label)	Returns the state indices for a specified label
`get_state_var_labels`()	Retruns a list of state variable labels
`set_state_var_label`(label, indices)	Adds a label for one or multiple state variabels in the state vector.
`trajectories`(xs, js)	Evaluation of the ALSSM for an array state vectors xs at evaluation indeces js.
`trajectory`(x, js)	Evaluation of the ALSSM for a state vector x at evaluation indeces js.
`update`()	Model update

Attributes

`A`	State matrix \(A \in \mathbb{R}^{N \times N}\)
`C`	Output matrix \(C \in \mathbb{R}^{L \times N}\)
`C_init`	Initialized Output matrix \(C \in \mathbb{R}^{L \times N}\)
`N`	Model order \(N\)
`alssms`	Set of models
`deltas`	Output scaling factors for each ALSSM in alssms
`gamma`	Decay factor per sample \(\gamma\)
`label`	Label of the model
`state_var_labels`	Dictionary containing state variable labels and index

property A#

State matrix \(A \in \mathbb{R}^{N \times N}\)

Type: ndarray, shape=(N, N)

property C#

Output matrix \(C \in \mathbb{R}^{L \times N}\)

Type: ndarray, shape=([L,] N)

property C_init#

Initialized Output matrix \(C \in \mathbb{R}^{L \times N}\)

Type: ndarray, shape=([L,] N)

property N#

Model order \(N\)

Type: int

property alssms#

Set of models

Type: list

property deltas#

Output scaling factors for each ALSSM in alssms

Type: np.ndarray

dump_tree()#

Returns the internal structure of the ALSSM model as a string.

Returns: out – String representing internal model structure.
Return type: str

Examples

>>> alssm_poly = lm.AlssmPoly(4, label="high order polynomial.rst")
>>> A = [[1, 1], [0, 1]]
>>> C = [[1, 0]]
>>> alssm_line = lm.Alssm(A, C, label="line")
>>> stacked_alssm = lm.AlssmStacked((alssm_poly, alssm_line), label='stacked model')
>>> print(stacked_alssm.dump_tree())
└-Alssm : stacked, A: (7, 7), C: (2, 7), label: stacked model
  └-Alssm : polynomial.rst, A: (5, 5), C: (1, 5), label: high order polynomial.rst
  └-Alssm : native, A: (2, 2), C: (1, 2), label: line

eval_state(x)#

Evaluation of the ALSSM for a state vector x.

eval_state(…) returns the ALSSM output

\[s_0(x) = CA^0x = Cx\]

for a state vector \(x\).

Parameters: x (array_like of shape=(N[,S])) – State vector \(x\)
Returns: s – ALSSM output
Return type: ndarray of shape=([L[,S]])

N : ALSSM system order, corresponding to the number of state variables
L : output order / number of signal channels
S : number of signal sets

Examples

>>> A = [[1, 1], [0, 1]]
>>> C = [1, 0]
>>> alssm = lm.Alssm(A, C, label='line')
>>>
>>> x = [0.1, 3]
>>> s = alssm.eval_state(x)
>>> print(s)
0.1

eval_states(xs)#

Evaluation of the ALSSM for an array of state vectors xs.

eval_states(…) returns the ALSSM output

\[s_0(x) = CA^0x = Cx\]

for each state vector \(x\) from the array xs

Parameters: xs (array_like of shape=(XS,N[,S])) – List of length XS with state vectors \(x\).
Returns: s – ALSSM outputs
Return type: ndarray of shape=(XS,[L[,S]])

N : ALSSM system order, corresponding to the number of state variables
L : output order / number of signal channels
S : number of signal sets

Examples

>>> A = [[1, 1], [0, 1]]
>>> C = [1, 0]
>>> alssm = lm.Alssm(A, C, label='line')
>>>
>>> xs = [[0.1, 3], [0, 1], [-0.8, 0.2], [1, -3]]
>>> s = alssm.eval_states(xs)
>>> print(s)
[ 0.1  0.  -0.8  1. ]

property gamma#

Decay factor per sample \(\gamma\)

Type: float

get_state_var_indices(label)#

Returns the state indices for a specified label

Parameters: label (str) – state label
Returns: out – state indices of the label
Return type: list of int

get_state_var_labels()#

Retruns a list of state variable labels

Returns: out – list of state variable labels
Return type: list

property label#

Label of the model

Type: str

set_state_var_label(label, indices)#

Adds a label for one or multiple state variabels in the state vector. Such labels are used to quickly referece to single states in the state vector by its names.

Parameters

label (str) – Label name
indices (tuple) – State indices

Examples

>>> alssm = lm.AlssmPoly(poly_degree=1, label='slope_with_offset')
>>> alssm.set_state_var_label('slope', (1,))
>>> alssm.state_var_labels
{'x': range(0, 2), 'x0': (0,), 'x1': (1,), 'slope': (1,)}
>>> alssm.state_var_labels['slope']
(1,)

property state_var_labels#

Dictionary containing state variable labels and index

Type: dict

trajectories(xs, js)#

Evaluation of the ALSSM for an array state vectors xs at evaluation indeces js.

trajectories(…) returns the ALSSM output

\[s_j(x) = CA^jx = Cx\]

for a state vector \(x\) and index \(j\) in the list js

Parameters

xs (array_like of shape=(XS,N[,S])) – List of length XS with state vectors \(x\).
js (array_like of shape=(J,)) – ALSSM evaluation indices

Returns

s – ALSSM outputs

Return type

ndarray of shape=(XS, J, [L[,S]])

N : ALSSM system order, corresponding to the number of state variables
L : output order / number of signal channels
S : number of signal sets
j : ALSSM evaluation index
J : number of ALSSM evaluation indices
XS : number of state vectors in a list

Examples

>>> A = [[1, 1], [0, 1]]
>>> C = [1, 0]
>>> alssm = lm.Alssm(A, C, label='line')
>>>
>>> xs = [[0.1, 3], [0, 1], [-0.8, 0.2], [1, -3]]
>>> s = alssm.trajectories(xs, js=[0, 1, 2, 3, 4, 5])
>>> print(s)

trajectory(x, js)#

Evaluation of the ALSSM for a state vector x at evaluation indeces js.

trajectory(…) returns the ALSSM output

\[s_j(x) = CA^jx = Cx\]

for a state vector \(x\) and index \(j\) in the list js

Parameters

x (array_like of shape=(N[,S])) – State vector \(x\)
js (array_like of shape=(J,)) – ALSSM evaluation indices

Returns

s – ALSSM outputs

Return type

ndarray of shape=(J, [L[,S]])

Examples

>>> A = [[1, 1], [0, 1]]
>>> C = [1, 0]
>>> alssm = lm.Alssm(A, C, label='line')
>>>
>>> x = [0.1, 3]
>>> s = alssm.trajectory(x, js=[0, 1, 2, 3, 4, 5])
>>> print(s)

update()#

Model update

Updates the internal model (A and C Matrix) based on the initialization parameters of a class.

lmlib.statespace.model.Alssm

lmlib.statespace.model.AlssmPoly