balredOptions

(Not recommended) Create option set for model order reduction

balred and balredOptions are not recommended. Use reducespec instead (since R2023b). For more information on updating your code, see Version History.

Syntax

opts = balredOptions
opts = balredOptions('OptionName', OptionValue)


Description

opts = balredOptions returns the default option set for the balred command.

opts = balredOptions('OptionName', OptionValue) accepts one or more comma-separated name/value pairs. Specify OptionName inside single quotes.

Name-Value Arguments

StateProjection	State elimination method. Specifies how to eliminate the weakly coupled states (states with smallest Hankel singular values). Specified as one of the following values: 'MatchDC' Discards the specified states and alters the remaining states to preserve the DC gain. 'Truncate' Discards the specified states without altering the remaining states. This method tends to product a better approximation in the frequency domain, but the DC gains are not guaranteed to match. Default: 'MatchDC'
ErrorBound	Error bound type, specified as either 'absolute' or 'relative'. Set 'Errorbound' to 'absolute' to control the absolute error $${\Vert G-G_r\Vert }_{\infty }$$. 'relative' to control the relative error $${\Vert G^{-1}\left(G-G_r\right)\Vert }_{\infty }$$. Relative error gives a better match across frequency while absolute error emphasizes areas with most gain. For more information, see getPeakGain. Default: 'absolute'
Regularization	Regularization level value that ensures a well-defined relative error at all frequencies. When you set ErrorBound to 'relative', balred reduces the model [sys,⍴*I] instead of sys. Set this option to 'auto' to let balred pick a suitable regularization level ⍴ value. Specify a value $$\rho \ge 0$$ to override this default. Default: 'auto'
FreqIntervals	Frequency intervals for computing frequency-limited Hankel singular values, 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. When identifying low-energy states to truncate, the software computes state contributions to system behavior in these frequency ranges only. 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. If StateProjection = 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified frequency intervals exclude 0. This behavior might reduce the quality of the match in the specified intervals. To improve the match within frequency intervals that exclude 0, set StateProjection = 'Truncate'. If both the frequency and time intervals do include DC, you can still set StateElimMethod = 'Truncate' to improve the match at other frequencies and times. Default: []
TimeIntervals	Time intervals for computing time-limited Hankel singular values, 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. When identifying low-energy states to truncate, 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. If StateProjection = 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified time intervals exclude Inf. This behavior might reduce the quality of the match in the specified intervals. To improve the match within time intervals that exclude Inf, set StateProjection = 'Truncate'. If both the frequency and time intervals do include DC, you can still set StateProjection = 'Truncate' to improve the match at other frequencies and times. Default: []
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. Increasing 'SepTol' helps separate nearby stable and unstable modes at the expense of accuracy. For more information, see stabsepOptions. Default: 10
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 to use them, see the balred reference page.

Examples

Reduced-Order Approximation with Offset Option

Compute a reduced-order approximation of the system given by:

$$G\left(s\right)=\frac{\left(s+0.5\right)\left(s+1.1\right)\left(s+2.9\right)}{\left(s+{10}^{-6}\right)\left(s+1\right)\left(s+2\right)\left(s+3\right)}.$$

Create the model.

sys = zpk([-0.5 -1.1 -2.9],[-1e-6 -2 -1 -3],1);

Exclude the pole at $$s=10^{-6}$$ from the stable term of the stable/unstable decomposition. To do so, set the Offset option of balredOptions to a value larger than the pole you want to exclude.

opts = balredOptions('Offset',0.001,'StateProjection','Truncate');

Visualize the Hankel singular values (HSV) and the approximation error.

balred(sys,opts)

The red HSV indicates that it is associated with an unstable mode.

Now, compute a second-order approximation with the specified options.

[rsys,info] = balred(sys,2,opts);
rsys

rsys =
 
  0.99113 (s+0.5235)
  -------------------
  (s+1e-06) (s+1.952)
 
Continuous-time zero/pole/gain model.

Notice that the pole at -1e-6 appears unchanged in the reduced model rsys.

Compare the responses of the original and reduced-order models.

bodeplot(sys,rsys,'r--')

Observe that the bode response of the original model and the reduced-order model nearly match.

Algorithms

The SepTol and Offset options are only used for models with unstable or marginally stable dynamics. Because Hankel singular values (HSV) are only meaningful for stable dynamics, balred must first split such models into the sum of their stable and unstable parts:

$$G=G_s+G_u$$

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

1. Large HSV 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. For instance, such loss of accuracy arises when trying to split a cluster of stable and unstable modes near s = 0. Because such clusters are numerically equivalent to multiple poles 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_u, unduly increasing its order. Such issues can be easily corrected by adjusting the SepTol tolerance.

If you use the TimeIntervals or FreqIntervals options, then balred 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].

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.

Version History

Introduced in R2010a

R2023b: Not recommended

balred and balredOptions are not recommended. To perform balanced truncation model order reduction, use the reducespec function with the following syntax.

R = reducespec(sys,"balanced");

For the full workflow, see Task-Based Model Order Reduction Workflow.

R2017b: MatchDC option honored when specified frequency or time intervals exclude DC

When you use balred for model reduction, you can use balredOptions to restrict the computation to specified frequency or time intervals. If the StateProjection option of balredOptions is set to 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified intervals exclude DC (frequency = 0 or time = Inf).

Prior to R2017b, if you specified time or frequency intervals that excluded DC, balred did not attempt to match the DC gain of the original and reduced models, even if StateProjection = 'MatchDC'.