`pymor.models.iosys`¶

Module Contents¶

Classes¶

`LTIModel`	Class for linear time-invariant systems.
`SecondOrderModel`	Class for linear second order systems.
`LinearDelayModel`	Class for linear delay systems.
`LinearStochasticModel`	Class for linear stochastic systems.
`BilinearModel`	Class for bilinear systems.

Functions¶

`sparse_min_size`	Return minimal sparse problem size for which to warn about converting to dense.
`_lti_to_poles_b_c`	Compute poles and residues.
`_poles_b_c_to_lti`	Create an `LTIModel` from poles and residue rank-1 factors.

pymor.models.iosys.sparse_min_size(value=1000)[source]¶: Return minimal sparse problem size for which to warn about converting to dense.

class pymor.models.iosys.LTIModel(A, B, C, D=None, E=None, cont_time=True, solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Bases: pymor.models.interface.Model

Class for linear time-invariant systems.

This class describes input-state-output systems given by

\[\begin{split}E(\mu) \dot{x}(t, \mu) & = A(\mu) x(t, \mu) + B(\mu) u(t), \\ y(t, \mu) & = C(\mu) x(t, \mu) + D(\mu) u(t),\end{split}\]

if continuous-time, or

\[\begin{split}E(\mu) x(k + 1, \mu) & = A(\mu) x(k, \mu) + B(\mu) u(k), \\ y(k, \mu) & = C(\mu) x(k, \mu) + D(\mu) u(k),\end{split}\]

if discrete-time, where \(A\), \(B\), \(C\), \(D\), and \(E\) are linear operators.

All methods related to the transfer function (e.g., frequency response calculation and Bode plots) are attached to the transfer_function attribute.

Parameters

A: The Operator A.
B: The Operator B.
C: The Operator C.
D: The Operator D or None (then D is assumed to be zero).
E: The Operator E or None (then E is assumed to be identity).
cont_time: True if the system is continuous-time, otherwise False.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

order[source]¶: The order of the system.

dim_input[source]¶: The number of inputs.

dim_output[source]¶: The number of outputs.

A[source]¶: The Operator A.

B[source]¶: The Operator B.

C[source]¶: The Operator C.

D[source]¶: The Operator D.

E[source]¶: The Operator E.

transfer_function[source]¶: The transfer function.

__str__(self)[source]¶: Return str(self).

classmethod from_matrices(cls, A, B, C, D=None, E=None, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create LTIModel from matrices.

Parameters

A: The NumPy array or SciPy spmatrix A.
B: The NumPy array or SciPy spmatrix B.
C: The NumPy array or SciPy spmatrix C.
D: The NumPy array or SciPy spmatrix D or None (then D is assumed to be zero).
E: The NumPy array or SciPy spmatrix E or None (then E is assumed to be identity).
cont_time: True if the system is continuous-time, otherwise False.
state_id: Id of the state space.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

lti: The LTIModel with operators A, B, C, D, and E.

to_matrices(self)[source]¶

Return operators as matrices.

Returns

A: The NumPy array or SciPy spmatrix A.
B: The NumPy array or SciPy spmatrix B.
C: The NumPy array or SciPy spmatrix C.
D: The NumPy array or SciPy spmatrix D or None (if D is a ZeroOperator).
E: The NumPy array or SciPy spmatrix E or None (if E is an IdentityOperator).

classmethod from_files(cls, A_file, B_file, C_file, D_file=None, E_file=None, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create LTIModel from matrices stored in separate files.

Parameters

A_file: The name of the file (with extension) containing A.
B_file: The name of the file (with extension) containing B.
C_file: The name of the file (with extension) containing C.
D_file: None or the name of the file (with extension) containing D.
E_file: None or the name of the file (with extension) containing E.
cont_time: True if the system is continuous-time, otherwise False.
state_id: Id of the state space.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

lti: The LTIModel with operators A, B, C, D, and E.

to_files(self, A_file, B_file, C_file, D_file=None, E_file=None)[source]¶

Write operators to files as matrices.

Parameters

A_file: The name of the file (with extension) containing A.
B_file: The name of the file (with extension) containing B.
C_file: The name of the file (with extension) containing C.
D_file: The name of the file (with extension) containing D or None if D is a ZeroOperator.
E_file: The name of the file (with extension) containing E or None if E is an IdentityOperator.

classmethod from_mat_file(cls, file_name, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create LTIModel from matrices stored in a .mat file.

Parameters

file_name: The name of the .mat file (extension .mat does not need to be included) containing A, B, C, and optionally D and E.
cont_time: True if the system is continuous-time, otherwise False.
state_id: Id of the state space.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

lti: The LTIModel with operators A, B, C, D, and E.

to_mat_file(self, file_name)[source]¶

Save operators as matrices to .mat file.

Parameters

file_name: The name of the .mat file (extension .mat does not need to be included).

classmethod from_abcde_files(cls, files_basename, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create LTIModel from matrices stored in .[ABCDE] files.

Parameters

files_basename: The basename of files containing A, B, C, and optionally D and E.
cont_time: True if the system is continuous-time, otherwise False.
state_id: Id of the state space.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

lti: The LTIModel with operators A, B, C, D, and E.

to_abcde_files(self, files_basename)[source]¶

Save operators as matrices to .[ABCDE] files in Matrix Market format.

Parameters

files_basename: The basename of files containing the operators.

__add__(self, other)[source]¶: Add an LTIModel.

__sub__(self, other)[source]¶: Subtract an LTIModel.

__neg__(self)[source]¶: Negate the LTIModel.

__mul__(self, other)[source]¶: Postmultiply by an LTIModel.

poles(self, mu=None)[source]¶

Compute system poles.

Note

Assumes the systems is small enough to use a dense eigenvalue solver.

Parameters

mu: Parameter values for which to compute the systems poles.

Returns

One-dimensional NumPy array of system poles.

eval_tf(self, s, mu=None)[source]¶

Evaluate the transfer function.

This function is an alias for transfer_function.eval_tf. See eval_tf for a detailed documentation.

eval_dtf(self, s, mu=None)[source]¶

Evaluate the derivative of the transfer function.

This function is an alias for transfer_function.eval_dtf. See eval_dtf for a detailed documentation.

freq_resp(self, w, mu=None)[source]¶

Evaluate the transfer function on the imaginary axis.

This function is an alias for transfer_function.freq_resp. See freq_resp for a detailed documentation.

bode(self, w, mu=None)[source]¶

Compute magnitudes and phases.

This function is an alias for transfer_function.bode. See bode for a detailed documentation.

bode_plot(self, w, mu=None, ax=None, Hz=False, dB=False, deg=True, **mpl_kwargs)[source]¶

Draw the Bode plot for all input-output pairs.

This function is an alias for transfer_function.bode_plot. See bode_plot for a detailed documentation.

mag_plot(self, w, mu=None, ax=None, ord=None, Hz=False, dB=False, **mpl_kwargs)[source]¶

Draw the magnitude plot.

This function is an alias for transfer_function.mag_plot. See mag_plot for a detailed documentation.

gramian(self, typ, mu=None)[source]¶

Compute a Gramian.

Parameters

typ

The type of the Gramian:

'c_lrcf': low-rank Cholesky factor of the controllability Gramian,
'o_lrcf': low-rank Cholesky factor of the observability Gramian,
'c_dense': dense controllability Gramian,
'o_dense': dense observability Gramian.

Note

For 'c_lrcf' and 'o_lrcf' types, the method assumes the system is asymptotically stable. For 'c_dense' and 'o_dense' types, the method assumes there are no two system poles which add to zero.

mu

Parameter values.

Returns

If typ is 'c_lrcf' or 'o_lrcf', then the Gramian factor as a VectorArray from self.A.source. If typ is 'c_dense' or 'o_dense', then the Gramian as a NumPy array.

_hsv_U_V(self, mu=None)[source]¶

Compute Hankel singular values and vectors.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

hsv: One-dimensional NumPy array of singular values.
Uh: NumPy array of left singular vectors.
Vh: NumPy array of right singular vectors.

hsv(self, mu=None)[source]¶

Hankel singular values.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

sv: One-dimensional NumPy array of singular values.

h2_norm(self, mu=None)[source]¶

Compute the H2-norm of the LTIModel.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

norm: H_2-norm.

hinf_norm(self, mu=None, return_fpeak=False, ab13dd_equilibrate=False)[source]¶

Compute the H_infinity-norm of the LTIModel.

Note

Assumes the system is asymptotically stable. Under this is assumption the H_infinity-norm is equal to the L_infinity-norm. Accordingly, this method calls linf_norm.

Parameters

mu: Parameter values.
return_fpeak: Whether to return the frequency at which the maximum is achieved.
ab13dd_equilibrate: Whether slycot.ab13dd should use equilibration.

Returns

norm: H_infinity-norm.
fpeak: Frequency at which the maximum is achieved (if return_fpeak is True).

hankel_norm(self, mu=None)[source]¶

Compute the Hankel-norm of the LTIModel.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

norm: Hankel-norm.

l2_norm(self, ast_pole_data=None, mu=None)[source]¶

Compute the L2-norm of the LTIModel.

The L2-norm of an LTIModel is defined via the integral

\[\lVert H \rVert_{\mathcal{L}_2} = \left( \frac{1}{2 \pi} \int_{-\infty}^{\infty} \lVert H(\boldsymbol{\imath} \omega) \rVert_{\operatorname{F}}^2 \operatorname{d}\!\omega \right)^{\frac{1}{2}}.\]

Parameters

ast_pole_data

Can be:

dictionary of parameters for eigs,
list of anti-stable eigenvalues (scalars),
tuple (lev, ew, rev) where ew contains the anti-stable eigenvalues and lev and rev are VectorArrays representing the eigenvectors.
None if anti-stable eigenvalues should be computed via dense methods.

mu

Parameter.

Returns

norm: L_2-norm.

linf_norm(self, mu=None, return_fpeak=False, ab13dd_equilibrate=False)[source]¶

Compute the L_infinity-norm of the LTIModel.

The L-infinity norm of an LTIModel is defined via

\[\lVert H \rVert_{\mathcal{L}_\infty} = \sup_{\omega \in \mathbb{R}} \lVert H(\boldsymbol{\imath} \omega) \rVert_2.\]

Parameters

mu: Parameter.
return_fpeak: Whether to return the frequency at which the maximum is achieved.
ab13dd_equilibrate: Whether slycot.ab13dd should use equilibration.

Returns

norm: L_infinity-norm.
fpeak: Frequency at which the maximum is achieved (if return_fpeak is True).

get_ast_spectrum(self, ast_pole_data=None, mu=None)[source]¶

Compute anti-stable subset of the poles of the LTIModel.

Parameters

ast_pole_data

Can be:

dictionary of parameters for eigs,
list of anti-stable eigenvalues (scalars),
tuple (lev, ew, rev) where ew contains the sorted anti-stable eigenvalues and lev and rev are VectorArrays representing the eigenvectors.
None if anti-stable eigenvalues should be computed via dense methods.

mu

Parameter.

Returns

lev: VectorArray of left eigenvectors.
ew: One-dimensional NumPy array of anti-stable eigenvalues sorted from smallest to largest.
rev: VectorArray of right eigenvectors.

class pymor.models.iosys.SecondOrderModel(M, E, K, B, Cp, Cv=None, D=None, cont_time=True, solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Bases: pymor.models.interface.Model

Class for linear second order systems.

This class describes input-output systems given by

\[\begin{split}M(\mu) \ddot{x}(t, \mu) + E(\mu) \dot{x}(t, \mu) + K(\mu) x(t, \mu) & = B(\mu) u(t), \\ y(t, \mu) & = C_p(\mu) x(t, \mu) + C_v(\mu) \dot{x}(t, \mu) + D(\mu) u(t),\end{split}\]

if continuous-time, or

\[\begin{split}M(\mu) x(k + 2, \mu) + E(\mu) x(k + 1, \mu) + K(\mu) x(k, \mu) & = B(\mu) u(k), \\ y(k, \mu) & = C_p(\mu) x(k, \mu) + C_v(\mu) x(k + 1, \mu) + D(\mu) u(k),\end{split}\]

if discrete-time, where \(M\), \(E\), \(K\), \(B\), \(C_p\), \(C_v\), and \(D\) are linear operators.

All methods related to the transfer function (e.g., frequency response calculation and Bode plots) are attached to the transfer_function attribute.

Parameters

M: The Operator M.
E: The Operator E.
K: The Operator K.
B: The Operator B.
Cp: The Operator Cp.
Cv: The Operator Cv or None (then Cv is assumed to be zero).
D: The Operator D or None (then D is assumed to be zero).
cont_time: True if the system is continuous-time, otherwise False.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

order[source]¶: The order of the system (equal to M.source.dim).

dim_input[source]¶: The number of inputs.

dim_output[source]¶: The number of outputs.

M[source]¶: The Operator M.

E[source]¶: The Operator E.

K[source]¶: The Operator K.

B[source]¶: The Operator B.

Cp[source]¶: The Operator Cp.

Cv[source]¶: The Operator Cv.

D[source]¶: The Operator D.

transfer_function[source]¶: The transfer function.

__str__(self)[source]¶: Return str(self).

classmethod from_matrices(cls, M, E, K, B, Cp, Cv=None, D=None, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create a second order system from matrices.

Parameters

M: The NumPy array or SciPy spmatrix M.
E: The NumPy array or SciPy spmatrix E.
K: The NumPy array or SciPy spmatrix K.
B: The NumPy array or SciPy spmatrix B.
Cp: The NumPy array or SciPy spmatrix Cp.
Cv: The NumPy array or SciPy spmatrix Cv or None (then Cv is assumed to be zero).
D: The NumPy array or SciPy spmatrix D or None (then D is assumed to be zero).
cont_time: True if the system is continuous-time, otherwise False.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

lti: The SecondOrderModel with operators M, E, K, B, Cp, Cv, and D.

to_matrices(self)[source]¶

Return operators as matrices.

Returns

M: The NumPy array or SciPy spmatrix M.
E: The NumPy array or SciPy spmatrix E.
K: The NumPy array or SciPy spmatrix K.
B: The NumPy array or SciPy spmatrix B.
Cp: The NumPy array or SciPy spmatrix Cp.
Cv: The NumPy array or SciPy spmatrix Cv or None (if Cv is a ZeroOperator).
D: The NumPy array or SciPy spmatrix D or None (if D is a ZeroOperator).

classmethod from_files(cls, M_file, E_file, K_file, B_file, Cp_file, Cv_file=None, D_file=None, cont_time=True, state_id='STATE', solver_options=None, error_estimator=None, visualizer=None, name=None)[source]¶

Create LTIModel from matrices stored in separate files.

Parameters

M_file: The name of the file (with extension) containing A.
E_file: The name of the file (with extension) containing E.
K_file: The name of the file (with extension) containing K.
B_file: The name of the file (with extension) containing B.
Cp_file: The name of the file (with extension) containing Cp.
Cv_file: None or the name of the file (with extension) containing Cv.
D_file: None or the name of the file (with extension) containing D.
cont_time: True if the system is continuous-time, otherwise False.
state_id: Id of the state space.
solver_options: The solver options to use to solve the Lyapunov equations.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

Returns

some: The SecondOrderModel with operators M, E, K, B, Cp, Cv, and D.

to_files(self, M_file, E_file, K_file, B_file, Cp_file, Cv_file=None, D_file=None)[source]¶

Write operators to files as matrices.

Parameters

M_file: The name of the file (with extension) containing M.
E_file: The name of the file (with extension) containing E.
K_file: The name of the file (with extension) containing K.
B_file: The name of the file (with extension) containing B.
Cp_file: The name of the file (with extension) containing Cp.
Cv_file: The name of the file (with extension) containing Cv or None if D is a ZeroOperator.
D_file: The name of the file (with extension) containing D or None if D is a ZeroOperator.

to_lti(self)[source]¶

Return a first order representation.

The first order representation

\[\begin{split}\begin{bmatrix} I & 0 \\ 0 & M \end{bmatrix} \frac{\mathrm{d}}{\mathrm{d}t}\! \begin{bmatrix} x(t) \\ \dot{x}(t) \end{bmatrix} & = \begin{bmatrix} 0 & I \\ -K & -E \end{bmatrix} \begin{bmatrix} x(t) \\ \dot{x}(t) \end{bmatrix} + \begin{bmatrix} 0 \\ B \end{bmatrix} u(t), \\ y(t) & = \begin{bmatrix} C_p & C_v \end{bmatrix} \begin{bmatrix} x(t) \\ \dot{x}(t) \end{bmatrix} + D u(t)\end{split}\]

is returned.

Returns

lti: LTIModel equivalent to the second-order model.

__add__(self, other)[source]¶: Add a SecondOrderModel or an LTIModel.

__radd__(self, other)[source]¶: Add to an LTIModel.

__sub__(self, other)[source]¶: Subtract a SecondOrderModel or an LTIModel.

__rsub__(self, other)[source]¶: Subtract from an LTIModel.

__neg__(self)[source]¶: Negate the SecondOrderModel.

__mul__(self, other)[source]¶: Postmultiply by a SecondOrderModel or an LTIModel.

__rmul__(self, other)[source]¶: Premultiply by an LTIModel.

poles(self, mu=None)[source]¶

Compute system poles.

Note

Assumes the systems is small enough to use a dense eigenvalue solver.

Parameters

mu: Parameter values.

Returns

One-dimensional NumPy array of system poles.

eval_tf(self, s, mu=None)[source]¶

Evaluate the transfer function.

This function is an alias for transfer_function.eval_tf. See eval_tf for a detailed documentation.

eval_dtf(self, s, mu=None)[source]¶

Evaluate the derivative of the transfer function.

This function is an alias for transfer_function.eval_dtf. See eval_dtf for a detailed documentation.

freq_resp(self, w, mu=None)[source]¶

Evaluate the transfer function on the imaginary axis.

This function is an alias for transfer_function.freq_resp. See freq_resp for a detailed documentation.

bode(self, w, mu=None)[source]¶

Compute magnitudes and phases.

This function is an alias for transfer_function.bode. See bode for a detailed documentation.

bode_plot(self, w, mu=None, ax=None, Hz=False, dB=False, deg=True, **mpl_kwargs)[source]¶

Draw the Bode plot for all input-output pairs.

This function is an alias for transfer_function.bode_plot. See bode_plot for a detailed documentation.

mag_plot(self, w, mu=None, ax=None, ord=None, Hz=False, dB=False, **mpl_kwargs)[source]¶

Draw the magnitude plot.

This function is an alias for transfer_function.mag_plot. See mag_plot for a detailed documentation.

gramian(self, typ, mu=None)[source]¶

Compute a second-order Gramian.

Parameters

typ

The type of the Gramian:

'pc_lrcf': low-rank Cholesky factor of the position controllability Gramian,
'vc_lrcf': low-rank Cholesky factor of the velocity controllability Gramian,
'po_lrcf': low-rank Cholesky factor of the position observability Gramian,
'vo_lrcf': low-rank Cholesky factor of the velocity observability Gramian,
'pc_dense': dense position controllability Gramian,
'vc_dense': dense velocity controllability Gramian,
'po_dense': dense position observability Gramian,
'vo_dense': dense velocity observability Gramian.

Note

For '*_lrcf' types, the method assumes the system is asymptotically stable. For '*_dense' types, the method assumes there are no two system poles which add to zero.

mu

Parameter values.

Returns

If typ is 'pc_lrcf', 'vc_lrcf', 'po_lrcf' or 'vo_lrcf', then the Gramian factor as a VectorArray from self.M.source. If typ is 'pc_dense', 'vc_dense', 'po_dense' or 'vo_dense', then the Gramian as a NumPy array.

psv(self, mu=None)[source]¶

Position singular values.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

One-dimensional NumPy array of singular values.

vsv(self, mu=None)[source]¶

Velocity singular values.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

One-dimensional NumPy array of singular values.

pvsv(self, mu=None)[source]¶

Position-velocity singular values.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

One-dimensional NumPy array of singular values.

vpsv(self, mu=None)[source]¶

Velocity-position singular values.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

One-dimensional NumPy array of singular values.

h2_norm(self, mu=None)[source]¶

Compute the H2-norm.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

norm: H_2-norm.

hinf_norm(self, mu=None, return_fpeak=False, ab13dd_equilibrate=False)[source]¶

Compute the H_infinity-norm.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.
return_fpeak: Should the frequency at which the maximum is achieved should be returned.
ab13dd_equilibrate: Should slycot.ab13dd use equilibration.

Returns

norm: H_infinity-norm.
fpeak: Frequency at which the maximum is achieved (if return_fpeak is True).

hankel_norm(self, mu=None)[source]¶

Compute the Hankel-norm.

Note

Assumes the system is asymptotically stable.

Parameters

mu: Parameter values.

Returns

norm: Hankel-norm.

class pymor.models.iosys.LinearDelayModel(A, Ad, tau, B, C, D=None, E=None, cont_time=True, error_estimator=None, visualizer=None, name=None)[source]¶

Bases: pymor.models.interface.Model

Class for linear delay systems.

This class describes input-state-output systems given by

\[\begin{split}E x'(t) & = A x(t) + \sum_{i = 1}^q{A_i x(t - \tau_i)} + B u(t), \\ y(t) & = C x(t) + D u(t),\end{split}\]

if continuous-time, or

\[\begin{split}E x(k + 1) & = A x(k) + \sum_{i = 1}^q{A_i x(k - \tau_i)} + B u(k), \\ y(k) & = C x(k) + D u(k),\end{split}\]

if discrete-time, where \(E\), \(A\), \(A_i\), \(B\), \(C\), and \(D\) are linear operators.

All methods related to the transfer function (e.g., frequency response calculation and Bode plots) are attached to the transfer_function attribute.

Parameters

A: The Operator A.
Ad: The tuple of Operators A_i.
tau: The tuple of delay times (positive floats or ints).
B: The Operator B.
C: The Operator C.
D: The Operator D or None (then D is assumed to be zero).
E: The Operator E or None (then E is assumed to be identity).
cont_time: True if the system is continuous-time, otherwise False.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

order[source]¶: The order of the system (equal to A.source.dim).

dim_input[source]¶: The number of inputs.

dim_output[source]¶: The number of outputs.

q[source]¶: The number of delay terms.

tau[source]¶: The tuple of delay times.

A[source]¶: The Operator A.

Ad[source]¶: The tuple of Operators A_i.

B[source]¶: The Operator B.

C[source]¶: The Operator C.

D[source]¶: The Operator D.

E[source]¶: The Operator E.

transfer_function[source]¶: The transfer function.

__str__(self)[source]¶: Return str(self).

__add__(self, other)[source]¶: Add an LTIModel, SecondOrderModel or LinearDelayModel.

__radd__(self, other)[source]¶: Add to an LTIModel or a SecondOrderModel.

__sub__(self, other)[source]¶: Subtract an LTIModel, SecondOrderModel or LinearDelayModel.

__rsub__(self, other)[source]¶: Subtract from an LTIModel or a SecondOrderModel.

__neg__(self)[source]¶: Negate the LinearDelayModel.

__mul__(self, other)[source]¶: Postmultiply an LTIModel, SecondOrderModel or LinearDelayModel.

__rmul__(self, other)[source]¶: Premultiply by an LTIModel or a SecondOrderModel.

eval_tf(self, s, mu=None)[source]¶

Evaluate the transfer function.

This function is an alias for transfer_function.eval_tf. See eval_tf for a detailed documentation.

eval_dtf(self, s, mu=None)[source]¶

Evaluate the derivative of the transfer function.

This function is an alias for transfer_function.eval_dtf. See eval_dtf for a detailed documentation.

freq_resp(self, w, mu=None)[source]¶

Evaluate the transfer function on the imaginary axis.

This function is an alias for transfer_function.freq_resp. See freq_resp for a detailed documentation.

bode(self, w, mu=None)[source]¶

Compute magnitudes and phases.

This function is an alias for transfer_function.bode. See bode for a detailed documentation.

bode_plot(self, w, mu=None, ax=None, Hz=False, dB=False, deg=True, **mpl_kwargs)[source]¶

Draw the Bode plot for all input-output pairs.

This function is an alias for transfer_function.bode_plot. See bode_plot for a detailed documentation.

mag_plot(self, w, mu=None, ax=None, ord=None, Hz=False, dB=False, **mpl_kwargs)[source]¶

Draw the magnitude plot.

This function is an alias for transfer_function.mag_plot. See mag_plot for a detailed documentation.

class pymor.models.iosys.LinearStochasticModel(A, As, B, C, D=None, E=None, cont_time=True, error_estimator=None, visualizer=None, name=None)[source]¶

Bases: pymor.models.interface.Model

Class for linear stochastic systems.

This class describes input-state-output systems given by

\[\begin{split}E \mathrm{d}x(t) & = A x(t) \mathrm{d}t + \sum_{i = 1}^q{A_i x(t) \mathrm{d}\omega_i(t)} + B u(t) \mathrm{d}t, \\ y(t) & = C x(t) + D u(t),\end{split}\]

if continuous-time, or

\[\begin{split}E x(k + 1) & = A x(k) + \sum_{i = 1}^q{A_i x(k) \omega_i(k)} + B u(k), \\ y(k) & = C x(k) + D u(t),\end{split}\]

if discrete-time, where \(E\), \(A\), \(A_i\), \(B\), \(C\), and \(D\) are linear operators and \(\omega_i\) are stochastic processes.

Parameters

A: The Operator A.
As: The tuple of Operators A_i.
B: The Operator B.
C: The Operator C.
D: The Operator D or None (then D is assumed to be zero).
E: The Operator E or None (then E is assumed to be identity).
cont_time: True if the system is continuous-time, otherwise False.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.

order[source]¶: The order of the system (equal to A.source.dim).

dim_input[source]¶: The number of inputs.

dim_output[source]¶: The number of outputs.

q[source]¶: The number of stochastic processes.

A[source]¶: The Operator A.

As[source]¶: The tuple of Operators A_i.

B[source]¶: The Operator B.

C[source]¶: The Operator C.

D[source]¶: The Operator D.

E[source]¶: The Operator E.

__str__(self)[source]¶: Return str(self).

class pymor.models.iosys.BilinearModel(A, N, B, C, D, E=None, cont_time=True, error_estimator=None, visualizer=None, name=None)[source]¶

Bases: pymor.models.interface.Model

Class for bilinear systems.

This class describes input-output systems given by

\[\begin{split}E x'(t) & = A x(t) + \sum_{i = 1}^m{N_i x(t) u_i(t)} + B u(t), \\ y(t) & = C x(t) + D u(t),\end{split}\]

if continuous-time, or

\[\begin{split}E x(k + 1) & = A x(k) + \sum_{i = 1}^m{N_i x(k) u_i(k)} + B u(k), \\ y(k) & = C x(k) + D u(t),\end{split}\]

if discrete-time, where \(E\), \(A\), \(N_i\), \(B\), \(C\), and \(D\) are linear operators and \(m\) is the number of inputs.

Parameters

A: The Operator A.
N: The tuple of Operators N_i.
B: The Operator B.
C: The Operator C.
D: The Operator D or None (then D is assumed to be zero).
E: The Operator E or None (then E is assumed to be identity).
cont_time: True if the system is continuous-time, otherwise False.
error_estimator: An error estimator for the problem. This can be any object with an estimate_error(U, mu, model) method. If error_estimator is not None, an estimate_error(U, mu) method is added to the model which will call error_estimator.estimate_error(U, mu, self).
visualizer: A visualizer for the problem. This can be any object with a visualize(U, model, ...) method. If visualizer is not None, a visualize(U, *args, **kwargs) method is added to the model which forwards its arguments to the visualizer’s visualize method.
name: Name of the system.