(Not recommended) Hankel singular values of dynamic system
hsv = hsvd(
hsv = hsvd(
[hsv,baldata] = hsvd(___)
computes the Hankel singular values
hsv = hsvd(
hsv of the dynamic system
sys. In state coordinates that equalize the input-to-state and state-to-output
energy transfers, the Hankel singular values measure the contribution of each state to the
input/output behavior. Hankel singular values are to model order what singular values are to
matrix rank. In particular, small Hankel singular values signal states that can be discarded to
simplify the model (see
For models with unstable poles,
hsvd only computes the Hankel singular
values of the stable part and entries of
hsv corresponding to unstable modes
are set to
computes the Hankel singular values using options that you specify using
hsv = hsvd(
hsvdOptions. Options include offset and tolerance options for computing the
stable-unstable decompositions. The options also allow you to limit the HSV computation to energy
contributions within particular time and frequency intervals. See
balredOptions for details.
[hsv,baldata] = hsvd(___) returns additional data to speed
up model order reduction. You can use this syntax with any of the previous combinations of input
hsvd(___) displays a Hankel singular values plot.
Compute Hankel Singular Values of System With Near-Unstable Pole
Create a system with a stable pole very near to 0, and display the Hankel singular values.
sys = zpk([1 2],[-1 -2 -3 -10 -1e-7],1); hsv = hsvd(sys)
hsv = 5×1 105 × 1.6667 0.0000 0.0000 0.0000 0.0000
Notice the dominant Hankel singular value with magnitude , which is so much larger that the significant digits of the other modes are not displayed. This value is due to the near-unstable mode at . Use the
'Offset' option to treat this mode as unstable.
opts = hsvdOptions('Offset',1e-7); hsvu = hsvd(sys,opts)
hsvu = 5×1 Inf 0.0688 0.0138 0.0024 0.0001
The Hankel singular value of modes that are unstable, or treated as unstable, is returned as
Inf. Create a Hankel singular-value plot while treating this mode as unstable.
ans = 5×1 Inf 0.0688 0.0138 0.0024 0.0001
The unstable mode is shown in red on the plot.
hsvd uses a linear scale. To switch the plot to a log scale, right-click on the plot and select Y Scale > Log. For information about programmatically changing properties of HSV plots, see
Frequency-Limited Hankel Singular Values
Compute the Hankel singular values of a model with low-frequency and high-frequency dynamics. Focus the calculation on the high-frequency modes.
Load the model and examine its frequency response.
load modeselect Gms bodeplot(Gms)
Gms has two sets of resonances, one at relatively low frequency and the other at relatively high frequency. Compute the Hankel singular values of the high-frequency modes, excluding the energy contributions to the low-frequency dynamics. To do so, use
hsvdOptions to specify a frequency interval above 30 rad/s.
opts = hsvdOptions('FreqInterval',[30 Inf]); hsvd(Gms,opts)
ans = 18×1 10-4 × 0.6237 0.4558 0.3183 0.2468 0.0895 0.0847 0.0243 0.0028 0.0000 0.0000 ⋮
To create a Hankel singular-value plot with more flexibility to programmatically customize
the plot, use
Version HistoryIntroduced before R2006a
hsvd is not recommended
Starting in R2021a, use the
balred command to compute Hankel singular values.
This approach facilitates better workflow since you can also compute reduced-order model
approximations using the same command. Furthermore, the enhanced workflow also includes new
options that preserve roll-off characteristics.
The following table shows some typical uses of
hsvd and how to update
your code to use
There are no plans to remove
hsvd at this time.