Hankel singular values of dynamic system


hsv = hsvd(sys)
hsv = hsvd(sys,'AbsTol',ATOL,'RelTol',RTOL,'Offset',ALPHA)
hsv = hsvd(sys, opts)
[hsv,baldata] = hsvd(sys)


hsv = hsvd(sys) computes the Hankel singular values 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.

hsv = hsvd(sys,'AbsTol',ATOL,'RelTol',RTOL,'Offset',ALPHA) specifies additional options for the stable/unstable decomposition. See the stabsep reference page for more information about these options. The default values are ATOL = 0, RTOL = 1e-8, and ALPHA = 1e-8.

hsv = hsvd(sys, opts) computes the Hankel singular values using the options specified in the hsvdOptions object opts.

hsvd(sys) displays a Hankel singular values plot.

[hsv,baldata] = hsvd(sys) returns additional data to speed up model order reduction with balred. For example

sys = rss(20);                 % 20-th order model
[hsv,baldata] = hsvd(sys);
rsys = balred(sys,8:10,'Balancing',baldata);

computes three approximations of sys of orders 8, 9, 10.

There is more than one hsvd available. Type

help lti/hsvd

for more information.


Compute Hankel Singular Values

This example illustrates how to compute Hankel singular values.

First, create a system with a stable pole very near to 0, then calculate the Hankel singular values.

sys = zpk([1 2],[-1 -2 -3 -10 -1e-7],1)

            (s-1) (s-2)
(s+1) (s+2) (s+3) (s+10) (s+1e-007)

For a better view of the Hankel singular values, switch the plot to log scale by selecting Y Scale > Log from the right-click menu.

Notice the dominant Hankel singular value with 1e5 magnitude, due to the mode s=-1e-7 near the imaginary axis. Set the offset=1e-6 to treat this mode as unstable


The dominant Hankel singular value is now shown as unstable.

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You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots.


The AbsTol, RelTol, and ALPHA parameters 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

  1. 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.

  2. 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.

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