Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-18 ni.com
Controller reduction proceeds by implementing the same connection rule
but on reduced versions of the two transfer function matrices.
When KE has been defined through Kalman filtering considerations, the
spectrum of the signal driving KE in Figure4-5 is white, with intensity Qyy.
It follows that to reflect in the multiple input case the different intensities
on the different scalar inputs, it is advisable to introduce at some stage a
weight into the reduction process.
Algorithm
After preliminary checks, the algorithm steps are:
1. Form the observability and weighted (through Qyy) controllability
grammians of E(s) in Equation 4-7 by
(4-8)
(4-9)
2. Compute the square roots of the eigenvalues of PQ (Hankel singular
values of the fractional representation of Equation 4-5). The maximum
order permitted is the number of nonzero eigenvalues of PQ that are
larger than ε.
3. Introduce the order of the reduced-order controller, possibly by
displaying the Hankel singular values (HSVs) to the user. Broadly
speaking, one can throw away small HSVs but not large ones.
4. Using redschur( )-type calculations, find a state-variable
description of Er(s). This means that Er(s) is the transfer function
matrix of a truncation of a balanced realization of E(s), but the
redschur( ) type calculations avoid the possibly numerically
difficult step of balancing the initially known realization of E(s).
Suppose that:
5. Define the reduced order controller Cr(s) by
(4-10)
so that
Qyy
12⁄
PA BK
R
–()′ABK
R
–()P+K–EQyyKE
′
=
QA BK
R
–()ABK
R
–()′Q+KR
′KRC′C––=
A
ˆSlbig
′ABK
R
–()Srbig KE
,Slbig
′KE
==
ACR Slbig
′ABK
R
–KEC–()Srbig
=
Crs() CCR sI ACR
–()
1–BCR
=