Propagation

For input units \([u]\), Propagation is the \(K\)-channel current-form propagation operator used by LineDistributed. It applies a fitted input factor, one scalar delay per mode, and a fitted output factor while preserving the input units.

\[\begin{aligned} \mathbf{H}(s) &= \sum_{m=1}^M \mathbf{H}^\mathrm{mps}_m(s) \exp(-s\tau_m) \end{aligned}\]

With \(\mathbf G_\mathrm{in}\) and \(\mathbf G_\mathrm{out}\) the transfer matrices of the input and output factors,

\[\mathbf{H}^\mathrm{mps}_m(s) = \mathbf{G}_\mathrm{out}(s)_{:,m}\,\mathbf{G}_\mathrm{in}(s)_{m,:}\]

Block Diagram

Figure 1: Propagation model

Model Parameters

Symbol	Units	JSON	Description	Note
\(K\)	[-]	K	Signal dimension	Required, positive integer

Parameter Validation

\[K \in \mathbb{Z}_{>0}\]

Model Derived Parameters

The full modal basis has one mode per channel:

\[\begin{split}\begin{aligned} M &= K \\ \boldsymbol{\tau} &= [\tau_1,\ldots,\tau_M]^\mathsf T \end{aligned}\end{split}\]

The modal delays are produced by the offline propagation fitting and enter through the delay bank’s delays coefficient set.

Model Ports

Symbol	Port	Type	Units	Description	Note
\(\mathbf{u}\)	input	Input	\([u]\)	Input vector port	\(\mathbf{u} \in \mathbb{R}^K\)
\(\mathbf{y}\)	out	Output	\([u]\)	Output vector port	\(\mathbf{y} \in \mathbb{R}^K\)

Submodels

Symbol	Description	Type	Order	JSON	Inputs	Outputs
\(\mathbf{g}_\mathrm{in}\)	Input factor	VectorFit	\(KQ_{\mathbf{g}_\mathrm{in}}\)	input	\(\mathbb{R}^K\)	\(\mathbb{R}^M\)
\(\mathbf{d}\)	Modal delay bank	Delay	History	delays	\(\mathbb{R}^M\)	\(\mathbb{R}^M\)
\(\mathbf{g}_\mathrm{out}\)	Output factor	VectorFit	\(KQ_{\mathbf{g}_\mathrm{out}}\)	output	\(\mathbb{R}^M\)	\(\mathbb{R}^K\)

Submodel Validation

Both rational factors must have stable poles and no term linear in \(s\). The input factor has algebraic input; the output factor has differential input from the modal delay bank.

\[\mathbf{E}^\mathrm{in}=\mathbf{E}^\mathrm{out}=\mathbf{0}\]

Submodel Wiring

\[\begin{split}\begin{aligned} \mathbf{w} &\leftarrow \mathbf{g}_\mathrm{in}[\mathbf{u}] \\ \mathbf{z} &\leftarrow \mathbf{d}[\mathbf{w}] \end{aligned}\end{split}\]

Model Variables

Internal Variables

Differential

None.

Algebraic

None.

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(\mathbf{u}\)	\([u]\)	Input vector	\(\mathbf{u} \in \mathbb{R}^K\)

Model Equations

Internal Equations

Differential

None.

Algebraic

None.

External Equations

\[\mathbf{y} \leftarrow \mathbf{g}_\mathrm{out}[\mathbf{z}]\]

Initialization

Initialize the rational factors using the EMT initialization contract. Provide the input-factor output history required by Delay.

Monitors

None.