Nomenclature, Nomenclature -6 – National Instruments NI MATRIXx Xmath User Manual
Page 13
Chapter 1
Introduction
1-6
ni.com
•
L
2
approximation, in which the L
2
norm of impulse response error (or,
by Parseval’s theorem, the L
2
norm of the transfer-function error along
the imaginary axis) serves as the error measure
•
Markov parameter or impulse response matching, moment matching,
covariance matching, and combinations of these, for example,
q-COVER approximation
•
Controller reduction using canonical interactions, balanced Riccati
equations, and certain balanced controller reduction algorithms
Nomenclature
This manual uses standard nomenclature. The user should be familiar with
the following:
•
sup denotes supremum, the least upper bound.
•
The acute accent (´) denotes matrix transposition.
•
A superscripted asterisk (*) denotes matrix transposition and complex
conjugation.
•
λ
max
(A) for a square matrix A denotes the maximum eigenvalue,
presuming there are no complex eigenvalues.
•
Re
λ
i
(A) and |
λ
i
(A)| for a square matrix A denote an arbitrary real part
and an arbitrary magnitude of an eigenvalue of A.
•
for a transfer function X(·) denotes:
•
An all-pass transfer-function W(s) is one where
for all
ω;
to each pole, there corresponds a zero which is the reflection through
the j
ω-axis of the pole, and there are no jω-axis poles.
•
An all-pass transfer-function matrix W(s) is a square matrix where
P > 0 and P
≥ 0 for a symmetric or hermitian matrix denote positive
and nonnegative definiteness.
•
P
1
> P
2
and P
1
≥ P
2
for symmetric or hermitian P
1
and P
2
denote
P
1
– P
2
is positive definite and nonnegative definite.
•
A superscripted number sign (#) for a square matrix A denotes the
Moore-Penrose pseudo-inverse of A.
X j
ω
( )
∞
sup
∞
–
ω ∞
< <
λ
max
X
*
j
ω
( )X jω
( )
[
]
[
]
1 2
/
X j
ω
( )
1
=
W
′ jω
–
(
)W jω
( )
I
=