kontrol.dmd package¶

Primary modules¶

kontrol.dmd.dmd module¶

Dynamic mode decomposition class

class kontrol.dmd.dmd.DMD(snapshot_1, snapshot_2=None, truncation_value=None, dt=1, run=False)¶

Bases: object

Dynamic mode decomposition class

Parameters:

snapshot_1 (array) – The 2-D snapshot array.
snapshot_2 (array, optional) – The 2-D snapshot array at next time step. If not provided, snapshot_1[:, :-1] becomes snapshot_1 and snapshot_2[:, 1:] becomes snapshot_2. Defaults None.
truncation_value (float, optional) – The truncation value (order/rank) of the system. Defaults None.
dt (float, optional) – The time difference between two snapshots. Defaults 1.
run (bool, optional) – Run dynamic mode decomposition upon construction. Computes DMD modes, Reduced-order model, etc. truncation_value must be specified for this to work. Defaults False.

snapshot_1¶

The 2-D snapshot matrix

Type:	array

snapshot_2¶

The 2-D snapshot matrix at next time step.

Type:	array

truncation_value¶

The truncation value (order/rank) of the system.

Type:	int

dt¶

The time difference between two snapshots.

Type:	float

u¶

The left singular vector of the snapshot_1 matrix.

Type:	array

vh¶

The right singular vector (conjugate-transposed) of the snapshot_1 matrix.

Type:	array

sigma¶

The singular values of the snapshot_1 matrix

Type:	array

u_truncated¶

The truncated left singular vector of the snapshot_1 matrix.

Type:	array

vh_truncated¶

The truncated right singular vector (conjugate-transposed) of the snapshot_1 matrix.

Type:	array

sigma_truncated¶

The truncated singular values of the snapshot_1 matrix

Type:	array

A_reduced¶

The reduced-order model of the dynamics.

Type:	array

w_reduced¶

The eigenvalues of the reduced-order model.

Type:	array

v_reduced¶

The eigenvectors of the reduced-order model.

Type:	array

dmd_modes¶

The DMD modes.

Type:	array

complex_frequencies¶

The complex frequencies of the modes.

Type:	array

v_constant¶

The constant vector of the time dynamics, determined by initial conditions.

Type:	array

time_dynamics¶

The time dynamics of the ODE solution.

Type:	array

prediction¶

The predicted time series.

Type:	array

A_reduced: Reduced-order model

complex_frequencies: Complex frequencies

compute_complex_frequencies(dt=None)¶

Compute the complex frequencies

Parameters:	dt (float, optional) – The time spacing between snapshots. Specified as self.dt or in constructor option. Defaults 1.
Returns:	complex_frequencies – Array of complex frequencies
Return type:	array

compute_dmd_modes()¶

Compute the DMD modes

Returns:	dmd_modes – The DMD modes.
Return type:	array

compute_reduced_model()¶

Compute the reduced-order model

Returns:	A_reduced – Matrix representing the reduced-order model.
Return type:	array

dt: Time difference between the snapshots

eig_reduced_model()¶

Eigen decomposition of the reduced-order model

Returns:	w_reduced (array) – The list of eigenvalues. v_reduced (array) – Eigenvalues as columns.

low_rank_approximation(truncation_value=None)¶

Truncate the svd.

truncation_value : int, optional: The truncation value (order/rank) of the system. Specify as self.truncation_value or via the constuctor option. Defaults None.

Returns:	u_truncated (array) – Truncated unitary matrix having left singular vectors as columns. sigma_truncated (array) – Truncated singular values. vh_truncated (array) – Truncated unitary matrix having right singular vectors as rows.

predict(t, t_constant=None, i_constant=0)¶

Predict the future states given a time array

Parameters:	t (array) – The time array t_constant (float, optional) – Time at which the constant vector is calculated. Defaults to be `t[0]` if not given. i_constant (int, optional) – Index of the `snapshot_1` at which the constant vector is calculated. Defaults 0.
Returns:	prediction – Predicted states.
Return type:	array

Notes

The constant vector is the vector that evolves and is evaluated using the initial/boundary values. Set t_constant and i_constant to where the prediction of the time series begins.

prediction: The ODE solution

run()¶: Run the DMD algorithm, compute dmd modes and complex frequencies

sigma: Singular values

sigma_truncated: Singular values truncated

snapshot_1: Snapshot 1

snapshot_2: Snapshot 1

svd()¶

Decompose the snapshot 1

Returns:	u (array) – Unitary matrix having left singular vectors as columns. sigma (array) – The singular values. vh (array) – Unitary matrix having right singular vectors as rows.

time_dynamics: Time dynamics of the ODE solution

truncation_value: Truncation value (order/rank) of the system

u: Left singular vectors

u_truncated: Left singular vectors (truncated)

v_constant: Constant vector of the ODE solution

vh: Right singular vectors

vh_truncated: Right singular vectors (truncated)

Secondary modules¶

kontrol.dmd.utils module¶

Utility functions for Dymanic Mode Decomposition.

kontrol.dmd.utils.auto_truncate(sigma, threshold=0.99)¶

Automatically get truncation value

Parameters:	sigma (array) – Singular values. Arranged from large to small values. threshold (float, optional) – Only include singular values so their sum is `threshold` of the original sum. Defaults 0.99.
Returns:	truncation_value – Amount of singular values that are required to reach the threshold sum.
Return type:	int

kontrol.dmd.utils.hankel(array, order)¶

Hankelize an array

Parameters:	array (array) – Array with rows as channels and columns as samples order (int) – The number of rows of the hankelized matrix.
Returns:	hankel_array – The hankelized array
Return type:	array

Examples