Specify Options for Frequency Response Fitting in Model Reducer

Since R2026a

Frequency grid

Frequencies for evaluating the frequency response of the original model, specified as a vector. Frequency values must be real, and can be positive or negative. This option is required for parametric models.

Specify a frequency grid that adequately captures gain and phase variations of the original model. In principle, stability is encoded in the frequency response. If the specified frequency grid is too sparse, the software may produce unstable fits. Therefore, you must provide a grid dense enough to accurately capture the system dynamics. For an example on how the choice of grid affects the fit quality, see Enforcing Stability in Fitted Reduced Models.

The software ignores this option for frd models.

Feedthrough

Feedthrough value of the fitted model, specified as a scalar, vector, or matrix compatible with model input-output size. When you specify a feedthrough value, the rational fit interpolates this value at s = Inf or z = Inf.

The software uses the following barycentric forms to fit the data to a rational approximation:

Standard form with scalar weights — $R (s) = \frac{f_{\infty} w_{\infty} + \sum_{j = 1}^{m} \frac{f_{j} w_{j}}{s - z_{j}}}{w_{\infty} + \sum_{j = 1}^{m} \frac{w_{j}}{s - z_{j}}} = \frac{N (s)}{D (s)}$
Here, f_∞ is the feedthrough value and w_∞ is the additional scalar-valued weight entry in the vector w for the feedthrough interpolation point. Additionally, z_j are the interpolation points, f_j are the corresponding values for f(z_j), and w_j are the scalar-valued weights.
MIMO form with matrix weights — $R (s) = (F_{\infty} W_{\infty} + \sum_{j = 1}^{m} \frac{F_{j} W_{j}}{s - z_{j}}) {(W_{\infty} + \sum_{j = 1}^{m} \frac{W_{j}}{s - z_{j}})}^{- 1} = N (s) D {(s)}^{- 1}$
Here, F_∞ is the feedthrough value, W_∞ is the additional matrix-valued weight entry in the vector W for the feedthrough interpolation point, and W_j are square matrices of size s. This software uses this form when Algorithm is set to BlockAAA when fitting a MIMO model of size r-by-s.

Algorithm

Fitting algorithm, specified as either "blockAAA" or "AAA".

For MIMO models, "blockAAA" uses matrix-valued weights while "AAA" uses scalar weights. "blockAAA" tends to yield lower-order models but is computationally more intensive. For SISO models, the algorithms are equivalent.

For "AAA", the software uses the barycentric form ([1]):
$R (s) = \frac{f_{\infty} w_{\infty} + \sum_{j = 1}^{m} \frac{f_{j} w_{j}}{s - z_{j}}}{w_{\infty} + \sum_{j = 1}^{m} \frac{w_{j}}{s - z_{j}}} = \frac{N (s)}{D (s)}$
Here, z_j are the interpolation points, f_j are the corresponding values for f(z_j), w_j are the scalar-valued weights, and f_∞ is the feedthrough value (if specified).
For "blockAAA", the software uses the barycentric form ([2]):
$R (s) = (F_{\infty} W_{\infty} + \sum_{j = 1}^{m} \frac{F_{j} W_{j}}{s - z_{j}}) {(W_{\infty} + \sum_{j = 1}^{m} \frac{W_{j}}{s - z_{j}})}^{- 1} = N (s) D {(s)}^{- 1}$
Here, W_j are square matrices of size s for MIMO models with size r-by-s.

Absolute error tolerance

Absolute error tolerance, specified as a nonnegative scalar.

For the specified frequencies w, the software ensures the fit error does not exceed AbsTol+RelTol×norm(F), where F is the frequency response at w.

The fit error is the maximum value of the normalized error E_k over the data points (z_k, F_k), where E_k is:

$E_{k} = \frac{{‖ F_{k} - F i t (z_{k}) ‖}_{F}}{(A b s T o l + R e l T o l \times {‖ F_{k} ‖}_{F})}$

The algorithm achieves the desired accuracy when this worst-case ratio drops below 1.

Relative error tolerance

Relative error tolerance, specified as a nonnegative scalar.