balred
Model order reduction
Description
[
computes a reduced-order approximation rsys
,info
] = balred(sys
,order
)rsys
of the LTI model
sys
. The desired order (number of states) is specified by
order
. You can try multiple orders at once by setting
order
to a vector of integers, in which case
rsys
is an array of reduced models. balred
also
returns a structure info
with additional information like the Hankel
singular values (HSV), error bound, regularization level and the Cholesky factors of the
gramians.
[~,
returns the structure info
] = balred(sys
)info
without computing the reduced-order model.
You can use this information to select the reduced order order
based
on your desired fidelity.
Note
When performance is a concern, avoid computing the Hankel singular values twice by
using the information obtained from the above syntax to select the desired model order
and then use rsys = balred(sys,order,info)
to compute the
reduced-order model.
[___] = balred(___,
computes the reduced model using the options set opts
)opts
that you
specify using balredOptions
. You can specify additional
options for eliminating states, using absolute vs. relative error control, emphasizing
certain time or frequency bands, and separating the stable and unstable modes. See
balredOptions
to create and configure the
option set opts
.
Examples
Input Arguments
Output Arguments
Algorithms
balred
first decomposes G into its stable and unstable parts:When you specify
ErrorBound
asabsolute
,balred
uses the balanced truncation method of [1] to reduce Gs. This computes the Hankel singular values (HSV) σj based on the controllability and observability gramians. For order r, the absolute error is bounded by . Here, n is the number of states in Gs.When you specify
ErrorBound
asrelative
,balred
uses the balanced stochastic truncation method of [2] to reduce Gs. For square Gs, this computes the HSV σj of the phase matrix where W(s) is a stable, minimum-phase spectral factor of GG’:For order r, the relative error is bounded by:
when, .
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.
[2] Green, M., "A Relative Error Bound for Balanced Stochastic Truncation", IEEE Transactions on Automatic Control, Vol. 33, No. 10, 1988