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Hankel singular values of dynamic system

* hsv* = hsvd(

`sys`

`hsv`

`sys`

`opts`

[hsv,baldata] = hsvd(___)

hsvd(___)

computes
the Hankel singular values * hsv* = hsvd(

`sys`

`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 `balred`

).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 `Inf`

.

computes
the Hankel singular values using options that you specify using * hsv* = hsvd(

`sys`

`opts`

`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 `hsvdOptions`

for details.`[hsv,baldata] = hsvd(___)`

returns
additional data to speed up model order reduction with `balred`

. You can use this syntax with
any of the previous combinations of input arguments.

`hsvd(___)`

displays a Hankel
singular values plot.

To create a Hankel singular-value plot with more flexibility
to programmatically customize the plot, use `hsvplot`

.

The `AbsTol`

, `RelTol`

, and `Offset`

options
of `hsvdOptions`

are only used for models with
unstable or marginally stable dynamics. Because Hankel singular values
are only meaningful for stable dynamics, `hsvd`

must
first split such models into the sum of their stable and unstable
parts:

G = G_s + G_ns

This decomposition can be tricky when the model has modes close
to the stability boundary (e.g., a pole at `s=-1e-10`

),
or clusters of modes on the stability boundary (e.g., double or triple
integrators). While `hsvd`

is able to overcome these
difficulties in most cases, it sometimes produces unexpected results
such as

Large Hankel singular values for the stable part.

This happens when the stable part

`G_s`

contains some poles very close to the stability boundary. To force such modes into the unstable group, increase the`'Offset'`

option to slightly grow the unstable region.Too many modes are labeled "unstable." For example, you see 5 red bars in the HSV plot when your model had only 2 unstable poles.

The stable/unstable decomposition algorithm has built-in accuracy checks that reject decompositions causing a significant loss of accuracy in the frequency response. Such loss of accuracy arises, e.g., when trying to split a cluster of stable and unstable modes near

`s=0`

. Because such clusters are numerically equivalent to a multiple pole at`s=0`

, it is actually desirable to treat the whole cluster as unstable. In some cases, however, large relative errors in low-gain frequency bands can trip the accuracy checks and lead to a rejection of valid decompositions. Additional modes are then absorbed into the unstable part`G_ns`

, unduly increasing its order.Such issues can be easily corrected by adjusting the

`AbsTol`

and`RelTol`

tolerances. By setting`AbsTol`

to a fraction of smallest gain of interest in your model, you tell the algorithm to ignore errors below a certain gain threshold. By increasing`RelTol`

, you tell the algorithm to sacrifice some relative model accuracy in exchange for keeping more modes in the stable part`G_s`

.

If you use the `TimeIntervals`

or `FreqIntervals`

options
of `hsvdOptions`

, then `hsvd`

bases
the computation of state energy contributions on time-limited or frequency-limited
controllability and observability Gramians. For information about
calculating time-limited and frequency-limited Gramians, see `gram`

and [1].

[1] Gawronski, W. and J.N. Juang. "Model
Reduction in Limited Time and Frequency Intervals." *International
Journal of Systems Science*. Vol. 21, Number 2, 1990, pp.
349–376.

`balreal`

| `balred`

| `hsvdOptions`

| `hsvplot`

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