lmlib.statespace.model.AlssmPoly#

class lmlib.statespace.model.AlssmPoly(poly_degree, C=None, **kwargs)#

ALSSM with discrete-time polynomial.rst output sequence

Representation of a Q-th degree polynomial.rst in i of the form

$P_Q(i) = x_0 i^0 + x_1 i^1 + ... + x_Q i^Q$

as an ALSSM, with transition matrix

$\begin{split}A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}\end{split}$

and state vector

$\begin{split}x = \begin{bmatrix} x_0 & x_1 & ... & x_N \\ \end{bmatrix}^T\end{split}$

For more details see [Wildhaber2019] PDF

Parameters
• poly_degree (int) – Polynomial degree Q. Corresponds to the highest exponent of the polynomial.rst. It follows a ALSSM system of order N = Q+1.

• C (array_like, shape=([L,] N), optional) – Output Matrix. If no output matrix is given, C gets initialize automatically to [1, 0, …] such that the shape is (N,). In addition with as_2dim_C=True C gets broadcated to shape (1, N). (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

Setting up a 4. order polynomial.rst, autonomous linear state space model.

>>> poly_degree = 3
>>> alssm = lm.AlssmPoly(poly_degree, label='poly')
>>> print(alssm)
A =
[[1 1 1 1]
[0 1 2 3]
[0 0 1 3]
[0 0 0 1]]
C =
[1 0 0 0]

>>> C = [[1, 0, 0], [0, 1, 0]]
>>> alssm = lm.AlssmPoly(poly_degree=2, C=C, label='poly')
>>> print(alssm)
A =
[[1 1 1]
[0 1 2]
[0 0 1]]
C =
[[1 0 0]
[0 1 0]]


Methods

 __init__(poly_degree[, C]) Returns the internal structure of the ALSSM model as a string. Evaluation of the ALSSM for a state vector x. Evaluation of the ALSSM for an array of state vectors xs. Returns the state indices for a specified label 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. 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 label Label of the model poly_degree Polynomial degree $$Q$$ (highest exponent/ order - 1) 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. ]

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

property poly_degree#

Polynomial degree $$Q$$ (highest exponent/ order - 1)

Type

int

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]])

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

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.