VectorFit

VectorFit represents a vector-fitted matrix rational approximation with real or complex poles and general residue matrices. For pole count \(Q\), define the pole-index set

\[\mathcal{Q} = \{q \in \mathbb{Z}_{>0} \mid q \le Q\}\]

\[\mathbf{H}(s) \approx \mathbf{D} + s\mathbf{E} + \sum_{q \in \mathcal{Q}} \dfrac{\mathbf{R}_q}{s - p_q}\]

VectorFit is a general rational operator; consuming models impose any physical constraints.

Note

The consuming model classifies the input \(\mathbf{u}\) as differential or algebraic. Algebraic input requires \(\mathbf{E} = \mathbf{0}\).

Block Diagram

Figure 1: VectorFit model

Model Parameters

For output dimension \(N\), input dimension \(K\), pole count \(Q\), input units \([u]\), and output units \([y]\):

Symbol	Units	JSON	Description	Note
\(\mathbf{D}\)	\([y]/[u]\)	D	Constant coefficient	\(\mathbf{D} \in \mathbb{R}^{N \times K}\)
\(\mathbf{E}\)	\(\mathrm{s}[y]/[u]\)	E	Linear coefficient	\(\mathbf{E} \in \mathbb{R}^{N \times K}\)
\(\mathbf{p}\)	\([\mathrm{s}^{-1}]\)	poles	Poles	\(\mathbf{p} \in \mathbb{C}^Q\)
\(\mathbf{R}\)	\([y]/(\mathrm{s}[u])\)	residues	Residues	\(\mathbf{R} \in \mathbb{C}^{N \times K \times Q}\)

Parameter Validation

Let \(\mathcal{Q}_\mathrm{r} \subseteq \mathcal{Q}\) contain the real-pole indices and \(\mathcal{Q}_\mathrm{c} \subseteq \mathcal{Q}\) the first indices of the nonreal conjugate pairs. Define their partner-index set as

\[\mathcal{Q}_\mathrm{c}^{+} = \{q+1 \mid q \in \mathcal{Q}_\mathrm{c}\}\]

The sets \(\mathcal{Q}_\mathrm{r}\), \(\mathcal{Q}_\mathrm{c}\), and \(\mathcal{Q}_\mathrm{c}^{+}\) are pairwise disjoint.

\[\begin{split}\begin{aligned} N &\in \mathbb{Z}_{>0} \\ K &\in \mathbb{Z}_{>0} \\ Q &\in \mathbb{Z}_{\ge 0} \\ \mathbf{E} &= \mathbf{0}, \quad \text{for algebraic input} \\ \mathcal{Q}_\mathrm{r} \cup \mathcal{Q}_\mathrm{c} \cup \mathcal{Q}_\mathrm{c}^{+} &= \mathcal{Q} \\ p_q &\in \mathbb{R}, \quad \mathbf{R}_q \in \mathbb{R}^{N \times K}, \quad q \in \mathcal{Q}_\mathrm{r} \\ p_{q+1} &= p_q^{\ast}, \quad q \in \mathcal{Q}_\mathrm{c} \\ \mathbf{R}_{q+1} &= \mathbf{R}_q^{\ast}, \quad q \in \mathcal{Q}_\mathrm{c} \end{aligned}\end{split}\]

Model Derived Parameters

\[\begin{split}\begin{aligned} \mathbf{a} &= \mathrm{Re}(\mathbf{p}) \\ \boldsymbol{\omega} &= \mathrm{Im}(\mathbf{p}) \\ \mathbf{A} &= \mathrm{Re}(\mathbf{R}) \\ \mathbf{B} &= \mathrm{Im}(\mathbf{R}) \end{aligned}\end{split}\]

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	\([y]\)	Output vector port	\(\mathbf{y} \in \mathbb{R}^N\)

Submodels

None.

Submodel Validation

None.

Submodel Wiring

None.

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{w}_q\)	\(\mathrm{s}[u]\)	Real memory states	\(\mathbf{w}_q \in \mathbb{R}^K\), \(q \in \mathcal{Q}_\mathrm{r} \cup \mathcal{Q}_\mathrm{c}\)
\(\mathbf{v}_q\)	\(\mathrm{s}[u]\)	Imaginary memory states	\(\mathbf{v}_q \in \mathbb{R}^K\), \(q \in \mathcal{Q}_\mathrm{c}\)

The documented real realization has order \(KQ\).

Algebraic

None.

External Variables

Differential

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

Algebraic

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

Model Equations

Internal Equations

Differential

\[\begin{split}\begin{aligned} 0 &= -\dfrac{\mathrm{d}\mathbf{w}_q}{\mathrm{d}t} + a_q\mathbf{w}_q + \mathbf{u}, \quad q \in \mathcal{Q}_\mathrm{r} \\ 0 &= -\dfrac{\mathrm{d}\mathbf{w}_q}{\mathrm{d}t} + a_q\mathbf{w}_q - \omega_q\mathbf{v}_q + \mathbf{u} \\ 0 &= -\dfrac{\mathrm{d}\mathbf{v}_q}{\mathrm{d}t} + \omega_q\mathbf{w}_q + a_q\mathbf{v}_q, \quad q \in \mathcal{Q}_\mathrm{c} \end{aligned}\end{split}\]

Algebraic

None.

External Equations

\[\mathbf{y} \leftarrow \mathbf{D}\mathbf{u} + \mathbf{E}\dfrac{\mathrm{d}\mathbf{u}}{\mathrm{d}t} + \sum_{q \in \mathcal{Q}_\mathrm{r}} \mathbf{A}_q\mathbf{w}_q + 2\sum_{q \in \mathcal{Q}_\mathrm{c}} (\mathbf{A}_q\mathbf{w}_q-\mathbf{B}_q\mathbf{v}_q)\]

Initialization

None.

Monitors

None.