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

$$\begin{array}{r}H=\left[\begin{array}{cccc}{h}_1& h_2& \dots & h_s\\ h_2& h_3& \dots & h_{s+1}\\ ⋮& ⋮& \ddots & ⋮\\ h_s& h_{s+1}& \dots & h_{2s-1}\end{array}\right]=U\mathrm{\Sigma }{V}^T\in {\mathbb{R}}^{ps\times ms},\end{array}$$

where $r\le min\{ms,ps\}$ 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 $n_L\le p$ and $n_R\le m$ 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 $ϵ$ is bounded by

$${ϵ}^2=\sum _{i=1}^{2s-1}\Vert C_r{A}_r^{i-1}{B}_r-h_i{\Vert }_F^2\le {\sigma }_{r+1}(\mathcal{H})\sqrt{r+p+m},$$

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

$${ϵ}^2=\sum _{i=1}^{2s-1}\Vert C_r{A}_r^{i-1}{B}_r-h_i{\Vert }_F^2\le 4(\sum _{i=n_L+1}^p{\sigma }_i^2({\mathrm{\Theta }}_L)+\sum _{i=n_R+1}^m{\sigma }_i^2({\mathrm{\Theta }}_R))+2{\sigma }_{r+1}(\mathcal{H})\sqrt{r+n_L+n_R},$$

where ${\mathrm{\Theta }}_L,{\mathrm{\Theta }}_R$ 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.