immrax.parametric

class immrax.parametric.Parametope(ox, alpha, y)

Bases: object

Parametope. Defines the set

\[{x : g(\alpha, x - \mathring{x}) <= y}\]

Attributes:

dtype

Methods

from_parametope(pt)
g(x)	Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x
tree_flatten()

tree_unflatten

property dtype: dtype

classmethod from_parametope(pt: Parametope)

g(x: Array | ndarray | bool | number | bool | int | float | complex)

Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x

Parameters:

alphaArrayLike

_description_

xArrayLike

_description_

tree_flatten()

classmethod tree_unflatten(aux_data, children)

ox: Array | ndarray | bool | number | bool | int | float | complex

alpha: Array | ndarray | bool | number | bool | int | float | complex

y: Array | ndarray | bool | number | bool | int | float | complex

class immrax.parametric.hParametope(ox, alpha, y)

Bases: Parametope

Defines a parametope with the particular structured nonlinearity

\[g(\alpha, x - \mathring{x}) = (-h(\alpha (x - \mathring{x})), h(\alpha (x - \mathring{x})))\]

and y split into lower and upper bounds y = (ly, uy).

Attributes:

dtype

Methods

from_parametope(pt)
g(x)	Evaluates the nonlinearity g at alpha, x
h(z)	Evaluates the nonlinearity h at z
hinv(iy)	Overapproximating inverse image of the nonlinearity h
k_face(k)	Overapproximate the k-face of the hParametope
tree_flatten()

tree_unflatten

classmethod from_parametope(pt: hParametope)

g(x: Array | ndarray | bool | number | bool | int | float | complex)

Evaluates the nonlinearity g at alpha, x

Parameters:

zArrayLike

Input to the nonlinearity

h(z: Array | ndarray | bool | number | bool | int | float | complex)

Evaluates the nonlinearity h at z

Parameters:

zArrayLike

Input to the nonlinearity

hinv(iy: Interval)

Overapproximating inverse image of the nonlinearity h

Parameters:

iyArrayLike

_description_

k_face(k: int) → Interval

Overapproximate the k-face of the hParametope

classmethod tree_unflatten(aux_data, children)

class immrax.parametric.ParametopeEmbedding(sys: System)

Bases: ABC

Methods

compute_reachset(t0, tf, pt0[, inputs, dt, ...])

compute_reachset(t0: Integer | Float, tf: Integer | Float, pt0: Parametope, inputs: List[Callable[[int, Array], Array]] = [], dt: float = 0.01, *, solver: Literal['euler', 'rk45', 'tsit5'] | AbstractSolver = 'tsit5', f_kwargs: immutabledict = immutabledict({}), **kwargs)

sys: System

class immrax.parametric.AdjointEmbedding(sys, alpha_p0, N0, kap: float = 0.1, permutation=None)

Bases: ParametopeEmbedding

Methods

compute_reachset(t0, tf, pt0[, inputs, dt, ...])

class immrax.parametric.FastlinAdjointEmbedding(sys, alpha_p0, N0, permutation=None, ustars=None, tt=None, kap=None)

Bases: ParametopeEmbedding

Methods

compute_reachset(t0, tf, pt0[, inputs, dt, ...])

class immrax.parametric.Ellipsoid(ox, alpha, y)

Bases: hParametope

Attributes:

P

dtype

Methods

g(x)	Evaluates the nonlinearity g at alpha, x
h(a)	Evaluates the nonlinearity h at z
hinv(y)	Overapproximating inverse image of the nonlinearity h
k_face(k)	Overapproximate the k-face of the hParametope
tree_flatten()

V
from_parametope
plot_projection
tree_unflatten

property P

V(x: Array | ndarray | bool | number | bool | int | float | complex)

classmethod from_parametope(pt: hParametope)

h(a: Array | ndarray | bool | number | bool | int | float | complex)

Evaluates the nonlinearity h at z

Parameters:

zArrayLike

Input to the nonlinearity

hinv(y)

Overapproximating inverse image of the nonlinearity h

Parameters:

iyArrayLike

_description_

plot_projection(ax, xi=0, yi=1, rescale=False, **kwargs)

class immrax.parametric.Polytope(ox, alpha, y)

Bases: hParametope

Attributes:

H

dtype

iy

ly

uy

Methods

from_parametope(pt)
g(x)	Evaluates the nonlinearity g at alpha, x
h(z)	Evaluates the nonlinearity h at z
hinv(y)	Overapproximating inverse image of the nonlinearity h
k_face(k)	Overapproximate the k-face of the hParametope
tree_flatten()

add_rows
from_interval
get_vertices
one_d_proj
plot_projection
tree_unflatten

property H

add_rows(Haug, Hp)

classmethod from_interval(*args)

classmethod from_parametope(pt: hParametope)

get_vertices()

h(z)

Evaluates the nonlinearity h at z

Parameters:

zArrayLike

Input to the nonlinearity

hinv(y)

Overapproximating inverse image of the nonlinearity h

Parameters:

iyArrayLike

_description_

property iy

property ly

one_d_proj(yi=0, rescale=False, **kwargs)

plot_projection(ax, xi=0, yi=1, rescale=False, **kwargs)

property uy

class immrax.parametric.Normotope(ox, alpha, y)

Bases: Parametope

Defines the set

\[{x : \|H(x - \ox)\| \leq y}\]

where \(\|\cdot\|\) is a norm, \(\ox\) is the center, \(H\) is a shaping matrix, and \(y\) is the offset.

Define \(h\) as the norm in subclasses, and \(\mu\) as the logarithmic norm associated to \(h\).

Attributes:

H

dtype

Methods

g(x)	Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x
h(z)	The norm associated to the normotope.
hinv(y)	An interval overapproximation of the inverse image of y under h.
induced_norm(A)	Computes the induced norm of A.
logarithmic_norm(A)	The logarithmic norm associated to h.
mu(A)	Alias for the logarithmic norm.
plot_projection(ax[, xi, yi, rescale])	Plot the projection of the normotope onto the xi-yi plane.
tree_flatten()
unvec(vec[, n])	Unvectorizes a vector into a normotope.
vec()	Vectorizes the normotope into a vector.

from_parametope
tree_unflatten

property H

classmethod from_parametope(pt: Parametope)

g(x)

Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x

Parameters:

alphaArrayLike

_description_

xArrayLike

_description_

h(z)

The norm associated to the normotope.

hinv(y)

An interval overapproximation of the inverse image of y under h.

classmethod induced_norm(A)

Computes the induced norm of A.

classmethod logarithmic_norm(A)

The logarithmic norm associated to h.

classmethod mu(A)

Alias for the logarithmic norm.

plot_projection(ax, xi=0, yi=1, rescale=False, **kwargs)

Plot the projection of the normotope onto the xi-yi plane.

classmethod unvec(vec, n=None)

Unvectorizes a vector into a normotope.

vec()

Vectorizes the normotope into a vector.

class immrax.parametric.LinfNormotope(ox, alpha, y)

Bases: Normotope

Defines the set

\[{x : \|H(x - \ox)\|_\infty \leq y}\]

Attributes:

H

dtype

Methods

g(x)	Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x
h(z)	The infinity norm
hinv(y)	An interval overapproximation of the inverse image of y under h.
induced_norm(A)	Computes the induced \(\ell_\infty\) norm of A
logarithmic_norm(A)	Computes the logarithmic \(\ell_\infty\) norm of A
mu(A)	Alias for the logarithmic norm.
plot_projection(ax[, xi, yi, rescale])	Plot the projection of the normotope onto the xi-yi plane.
tree_flatten()
unvec(vec[, n])	Unvectorizes a vector into a normotope.
vec()	Vectorizes the normotope into a vector.

from_interval
from_normotope
from_parametope
to_polytope
tree_unflatten

classmethod from_interval(*args)

classmethod from_normotope(nt: Normotope)

h(z)

The infinity norm

hinv(y)

An interval overapproximation of the inverse image of y under h.

classmethod induced_norm(A)

Computes the induced \(\ell_\infty\) norm of A

classmethod logarithmic_norm(A)

Computes the logarithmic \(\ell_\infty\) norm of A

plot_projection(ax, xi=0, yi=1, rescale=False, **kwargs)

Plot the projection of the normotope onto the xi-yi plane.

to_polytope() → Polytope

class immrax.parametric.L2Normotope(ox, alpha, y)

Bases: Normotope

Defines the set

\[{x : \|H(x - \ox)\|_2 \leq y}\]

Attributes:

H

dtype

Methods

g(x)	Evaluates the nonlinearity \(g(\alpha, x - \mathring{x})\) at x
h(z)	The L_2 norm
hinv(y)	An interval overapproximation of the inverse image of y under h.
induced_norm(A)	Computes the induced \(\ell_2\) norm of A
logarithmic_norm(A)	Computes the \(\ell_2\) logarithmic norm of A
mu(A)	Alias for the logarithmic norm.
plot_projection(ax[, xi, yi, rescale])	Plot the projection of the normotope onto the xi-yi plane.
tree_flatten()
unvec(vec[, n])	Unvectorizes a vector into a normotope.
vec()	Vectorizes the normotope into a vector.

from_interval
from_normotope
from_parametope
tree_unflatten

classmethod from_interval(*args)

classmethod from_normotope(nt: Normotope)

h(z)

The L_2 norm

hinv(y)

An interval overapproximation of the inverse image of y under h.

classmethod induced_norm(A)

Computes the induced \(\ell_2\) norm of A

classmethod logarithmic_norm(A)

Computes the \(\ell_2\) logarithmic norm of A

plot_projection(ax, xi=0, yi=1, rescale=False, **kwargs)

Plot the projection of the normotope onto the xi-yi plane.