Chapter 1 Introduction
Xmath Model Reduction Module 1-14 ni.com
nonnegative hermitian for all ω. If Φ is scalar, then Φ(jω)≥0 for all ω.
Normally one restricts attention to Φ(·) with limω→∞Φ(jω)<∞. A key result
is that, given a rational, nonnegative hermitian Φ(jω) with
limω→∞Φ(jω)<∞, there exists a rational W(s) where,
•W(∞)<∞.
•W(s) is stable.
•W(s) is minimum phase, that is, the rank of W(s) is constant in Re[s]>0.
In the scalar case, all zeros of W(s) lie in Re[s]≤0, or in Re[s]<0 if Φ(jω)>0
for all ω.
In the matrix case, and if Φ(jω) is nonsingular for some ω, it means that
W(s) is square and W–1(s) has all its poles in Re[s]≤ 0, or in Re[s]<0 if Φ(jω)
is nonsingular for all ω.
Moreover, the particular W(s) previously defined is unique, to within right
multiplication by a constant orthogonal matrix. In the scalar case, this
means that W(s) is determined to within a ±1 multiplier.
Example 1-1 Example of Spectral Factorization
Suppose:
Then Equation1 -3 is satisfied by , which is stable and
minimum phase.
Also, Equation1-3 is satisfied by and , and , and
so forth, but none of these is minimum phase.
bst( ) and mulhank( ) both require execution within the program of
a spectral factorization; the actual algorithm for achieving the spectral
factorization depends on a Riccati equation. The concepts of a spectrum
and spectral factor also underpin aspects of wtbalance( ).
Φjω() ω21+
ω24+
---------------=
Ws() s1+
s2+
-----------
±=
s1–
s2+
----------- s3–
s2+
----------- s1–
s2+
----------- esT–s1+
s2+
-----------