Chapter 1 Introduction
Xmath Model Reduction Module 1-6 ni.com
•L2 approximation, in which the L2 norm of impulse response error (or,
by Parseval’s theorem, the L2 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.
•λmax(A) for a square matrix A denotes the maximum eigenvalue,
presuming there are no complex eigenvalues.
•Reλi(A) and |λi(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 jω-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 sy mmetric or hermitian matrix denote positive
and nonnegative definiteness.
•P1>P2 and P1≥P2 for symmetric or hermitian P1 and P2 denote
P1–P2 is positive definite and nonnegative definite.
•A superscripted number sign (#) for a square matrix A denotes the
Moore-Penrose pseudo-inverse of A.
Xjω()
∞
sup
∞–ω∞<<λmax X*jω()Xjω()[][]
12/
Xjω() 1=
W′jω–()Wjω()I=