National Instruments Xmath Hankel Norm Approximation

Chapter 2 Additive Error Reduction

Xmath Model Reduction Module 2-6 ni.com

with controllability and observability grammians given by,

in which the diagonal entries of Σ are in decreasing order, that is,

σ1≥σ

2≥···, and such that the last diagonal entry of Σ1 exceeds

the first diagonal entry of Σ2. It turns out that Reλi()<0 and

Reλi(A11–A12 A21)< 0, and a reduced order model Gr(s) can be

definedby:

The attractive feature [LiA89] is that the same error bound holds as for

balanced truncation. For example,

Although the error bounds are the same, the actual frequency pattern of

the e rrors, and th e actual maximum modulus, need not be the same for

reduction to the same order. One crucial difference is that balanced

truncation provides exact matching at ω=∞, but does not match at DC,

while singular perturbation is exactly the other way round. Perfect

matching at DC can be a substantial advantage, especially if input signals

are known to be band-limited.

Singular perturbation can be achieved with mreduce( ). Figure2-1 shows

the two alternative approaches. For both continuous-time and discrete-time

reductions, the end result is a balanced realization.

Hankel Norm Approximation

In Hankel norm approximation, one relies on the fact that if one chooses an

approximation to exactly minimize one norm (the Hankel norm) then the

infinity norm will be approximately minimized. The Hankel norm is

defined in the following way. Let G(s) be a (rational) stable transfer

PQΣΣ10

0Σ2

===

A22

1–

A22

1–

·A11 A12A22

1–A21

–()xB

1A12

–A22

1–B2

+()u+=

1C2A22

1–A21

–()xDC

2A22

1–B2

–()u+=

Gjω()Grjω()–∞2trΣ2

≤