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hsvdOptions

(Not recommended) Create option set for computing Hankel singular values and input/output balancing

hsvdOptions is not recommended. Use balredOptions instead. For more information, see Compatibility Considerations.

Syntax

opts = hsvdOptions
opts = hsvdOptions(Name,Value)

Description

opts = hsvdOptions returns the default options for the hsvd and balreal commands.

opts = hsvdOptions(Name,Value) returns an options set with the options specified by one or more Name,Value pair arguments.

Input Arguments

Name-Value Pair Arguments

Specify comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'FreqIntervals'

Frequency intervals for computing frequency-limited Hankel singular values (balred) or balanced realization (balreal), specified as a matrix with two columns. Each row specifies a frequency interval [fmin fmax], where fmin and fmax are nonnegative frequencies, expressed in the frequency unit of the model. For example:

  • To restrict the computation to the range between 3 rad/s and 15 rad/s, assuming the frequency unit of the model is rad/s, set FreqIntervals to [3 15].

  • To restrict the computation to two frequency intervals, 3-15 rad/s and 40-60 rad/s, use [3 15; 40 60].

  • To specify all frequencies below a cutoff frequency fcut, use [0 fcut].

  • To specify all frequencies above the cutoff, use [fcut Inf] in continuous time, or [fcut pi/Ts] in discrete time, where Ts is the sample time of the model.

The default value, [], imposes no frequency limitation and is equivalent to [0 Inf] in continuous time or [0 pi/Ts] in discrete time. However, if you specify a TimeIntervals value other than [], then this limit overrides FreqIntervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

Default: []

'TimeIntervals'

Time intervals for computing time-limited Hankel singular values (balred) or balanced realization (balreal), specified as a matrix with two columns. Each row specifies a time interval [tmin tmax], where tmin and tmax are nonnegative times, expressed in the time unit of the model. The software computes state contributions to the system’s impulse response in these time intervals only. For example:

  • To restrict the computation to the range between 3 s and 15 s, assuming the time unit of the model is seconds, set TimeIntervals to [3 15].

  • To restrict the computation to two time intervals, 3-15 s and 40-60 s, use [3 15; 40 60].

  • To specify all times from zero up to a cutoff time tcut, use [0 tcut]. To specify all times after the cutoff, use [tcut Inf].

The default value, [], imposes no time limitation and is equivalent to [0 Inf]. However, if you specify a FreqIntervals value other than [], then this limit overrides Timeintervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

'SepTol'

Maximum loss of accuracy value in stable and unstable decomposition. For models with unstable poles, balred first extracts the stable dynamics using stabsep. Use 'SepTol' to control the decomposition accuracy.

For more information, see balredOptions.

Default: SepTol = 'auto'

'Offset'

Offset for the stable/unstable boundary. Positive scalar value. In the stable/unstable decomposition, the stable term includes only poles satisfying:

  • Re(s) < -Offset * max(1,|Im(s)|) (Continuous time)

  • |z| < 1 - Offset (Discrete time)

Increase the value of Offset to treat poles close to the stability boundary as unstable.

Default: 1e-8

For additional information on the options and how they affect the calculation, see balredOptions. The time-limited and frequency-limited state contributions are calculated using the time-limited and frequency-limited controllability and observability Gramians, as described in gram and in [1].

Examples

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Compute the Hankel singular values of the system given by:

sys=(s+0.5)(s+10-6)(s+2)

Use the Offset option to force hsvd to exclude the pole at s=10-6 from the stable term of the stable/unstable decomposition.

sys = zpk(-.5,[-1e-6 -2],1); 
opts = hsvdOptions('Offset',.001); 
hsvd(sys,opts)
ans = 2×1

       Inf
    0.1875

The plot shows that one state is treated as unstable. hsvd computes the energy contributions of the stable states only.

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)

Figure contains 2 axes. Axes 1 contains an object of type line. This object represents Gms. Axes 2 contains an object of type line. This object represents 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
      ⋮

Compatibility Considerations

expand all

Not recommended starting in R2021a

References

[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.

Introduced in R2010a