balred
Model order reduction
Syntax
rsys = balred(sys,ORDERS)
rsys = balred(sys,ORDERS,BALDATA)
rsys = balred(___,opts)
Description
rsys = balred(sys,ORDERS)rsys of the LTI model sys.
The desired order (number of states) for rsys is
specified by ORDERS. You can try multiple orders
at once by setting ORDERS to a vector of integers,
in which case rsys is a vector of reduced-order
models. balred uses implicit balancing techniques
to compute the reduced- order approximation rsys.
Use hsvd to plot the Hankel
singular values and pick an adequate approximation order. States with
relatively small Hankel singular values can be safely discarded.
When sys has unstable poles, it is first
decomposed into its stable and unstable parts using stabsep, and only the stable part is
approximated. Use balredOptions to
specify additional options for the stable/unstable decomposition.
When you have System Identification Toolbox™ software
installed, sys can only be an identified state-space
model (idss). The reduced-order model is also an idss model.
rsys = balred(sys,ORDERS,BALDATA)hsvd.
Because hsvd does most of the work needed to compute rsys,
this syntax is more efficient when using hsvd and balred jointly.
rsys = balred(___,opts)balredOptions. Options include offset
and tolerance options for computing the stable-unstable decompositions.
There also options for emphasizing particular time or frequency intervals.
See balredOptions for details.
Note
The order of the approximate model is always at least the number
of unstable poles and at most the minimal order of the original model
(number NNZ of nonzero Hankel singular values using
an eps-level relative threshold)
Examples
Reduced-Order Approximation with Offset Option
Compute a reduced-order approximation of the system given by:
Create the model.
sys = zpk([-.5 -1.1 -2.9],[-1e-6 -2 -1 -3],1);
Exclude the pole at 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.
opt = balredOptions('Offset',.001,'StateElimMethod','Truncate');
Compute a second-order approximation.
rsys = balred(sys,2,opt);
Compare the responses of the original and reduced-order models.
bodeplot(sys,rsys,'r--')
Model Reduction in a Particular Frequency Band
Reduce a high-order model with a focus on the dynamics in a particular frequency range.
Load a model and examine its frequency response.
load('highOrderModel.mat','G') bodeplot(G)
G is a 48th-order model with several large peak regions around 5.2 rad/s, 13.5 rad/s, and 24.5 rad/s, and smaller peaks scattered across many frequencies. Suppose that for your application you are only interested in the dynamics near the second large peak, between 10 rad/s and 22 rad/s. Focus the model reduction on the region of interest to obtain a good match with a low-order approximation. Use balredOptions to specify the frequency interval for balred.
bopt = balredOptions('StateElimMethod','Truncate','FreqIntervals',[10,22]); GLim10 = balred(G,10,bopt); GLim18 = balred(G,18,bopt);
Examine the frequency responses of the reduced-order models. Also, examine the difference between those responses and the original response (the absolute error).
subplot(2,1,1); bodemag(G,GLim10,GLim18,logspace(0.5,1.5,100)); title('Bode Magnitude Plot') legend('Original','Order 10','Order 18'); subplot(2,1,2); bodemag(G-GLim10,G-GLim18,logspace(0.5,1.5,100)); title('Absolute Error Plot') legend('Order 10','Order 18');
With the frequency-limited energy computation, even the 10th-order approximation is quite good in the region of interest.
Compatibility Considerations
References
[1] Varga, A., "Balancing-Free Square-Root Algorithm for Computing Singular Perturbation Approximations," Proc. of 30th IEEE CDC, Brighton, UK (1991), pp. 1062-1065.