pymor.reductors.era

Module Contents

class pymor.reductors.era.ERAReductor(data, sampling_time, force_stability=True, feedthrough=None)

Bases: pymor.core.cache.CacheableObject

Eigensystem Realization Algorithm reductor.

Constructs a (reduced) realization from a sequence of Markov parameters \(h_i\), for \(i\in\{1,\,\dots,\,2s-1\}\), \(s\in\mathbb{N}\), by a (reduced) orthogonal factorization of the Hankel matrix of Markov parameters

\[\begin{split}H = \begin{bmatrix} h_1 & h_2 & \dots & h_s \\ h_2 & h_3 & \dots & h_{s+1}\\ \vdots & \vdots & \ddots & \vdots\\ h_s & h_{s+1} & \dots & h_{2s-1} \end{bmatrix}=U\Sigma V^T\in\mathbb{R}^{ps\times ms},\end{split}\]

where \(r\leq\min\{ms,ps\}\) is the reduced order. See [Kun78].

In order for the identified model to be stable, the Markov parameters decay substantially within \(s\) samples. Stability is enforced automatically through zero-padding and can be deactivated by setting force_stability=False.

For a large number of inputs and/or outputs, the factorization of the Hankel matrix can be accelerated by tangentially projecting the Markov parameters to reduce the dimension of the Hankel matrix, i.e.

\[\hat{h}_i = W_L^T h_i W_R,\]

where \(n_L \leq p\) and \(n_R \leq m\) are the number of left and right tangential directions and \(W_L \in \mathbb{R}^{p \times n_L}\) an \(W_R \in \mathbb{R}^{m \times n_R}\) are the left and right projectors, respectively. See [KG16].

data

NumPy array that contains the first \(n\) Markov parameters of an LTI system. Has to be one- or three-dimensional with either:

data.shape == (n,)

for scalar-valued Markov parameters or:

data.shape == (n, p, m)

for matrix-valued Markov parameters of dimension \(p\times m\), where \(m\) is the number of inputs and \(p\) is the number of outputs of the system.

sampling_time

A number that denotes the sampling time of the system (in seconds).

force_stability

Whether the Markov parameters are zero-padded to double the length in order to enforce Kung’s stability assumption. See [Kun78]. Defaults to True.

feedthrough

(Optional) Operator or NumPy array of shape (p, m). The zeroth Markov parameter that defines the feedthrough of the realization. Defaults to None.

Methods

error_bounds	Compute the error bounds for all possible reduction orders.
input_projector	Construct the right/input projector \(W_2\).
output_projector	Construct the left/output projector \(W_1\).
reduce	Construct a minimal realization.

cache_region = 'memory'

error_bounds(num_left=None, num_right=None)

Compute the error bounds for all possible reduction orders.

Without tangential projection of the Markov parameters, the \(\mathcal{L}_2\)-error of the Markov parameters \(\epsilon\) is bounded by

\[\epsilon^2 = \sum_{i = 1}^{2 s - 1} \lVert C_r A_r^{i - 1} B_r - h_i \rVert_F^2 \leq \sigma_{r + 1}(\mathcal{H}) \sqrt{r + p + m},\]

where \((A_r,B_r,C_r)\) is the reduced realization of order \(r\), \(h_i\in\mathbb{R}^{p\times m}\) is the \(i\)-th Markov parameter and \(\sigma_{r+1}(\mathcal{H})\) is the first neglected singular value of the Hankel matrix of Markov parameters.

With tangential projection, the bound is given by

\[\epsilon^2 = \sum_{i = 1}^{2 s - 1} \lVert C_r A_r^{i - 1} B_r - h_i \rVert_F^2 \leq 4 \left( \sum_{i = n_L + 1}^p \sigma_i^2(\Theta_L) + \sum_{i = n_R + 1}^m \sigma_i^2(\Theta_R) \right) + 2 \sigma_{r + 1}(\mathcal{H}) \sqrt{r + n_L + n_R},\]

where \(\Theta_L,\,\Theta_R\) is the matrix of horizontally or vertically stacked Markov parameters, respectively. See [KG16] (Thm. 3.4) for details.

input_projector(num_right)

Construct the right/input projector \(W_2\).

output_projector(num_left)

Construct the left/output projector \(W_1\).

reduce(r=None, tol=None, num_left=None, num_right=None)

Construct a minimal realization.

Parameters

r

Order of the reduced model if tol is None, maximum order if tol is specified.

tol

Tolerance for the error bound.

num_left

Number of left (output) directions for tangential projection.

num_right

Number of right (input) directions for tangential projection.

Returns

rom

Reduced-order LTIModel.