National Instruments NI MATRIXx Xmath - Manual (Page 14)

Chapter 1 Introduction
© National Instruments Corporation 1-7 Xmath Model Reduction Module
• An inequality or bound is tight if it can be met in practice, for example
is tight because the inequality becomes an equality for x = 1. Again,
if F(j
ω) denotes the Fourier transform of some
, the
Heisenberg inequality states,
and the bound is tight since it is attained for f(t) = exp + (–kt 2 ).
Commonly Used Concepts
This section outlines some frequently used standard concepts.
Controllability and Observability Grammians
Suppose that G(s) = D + C(sI–A) –1 B is a transfer-function matrix with
Re
λ
i
(A)<0. Then there exist symmetric matrices P, Q defined by:
PA
′ + AP = –BB′
QA + A
′Q = –C′C
These are termed the controllability and observability grammians of the
realization defined by {A,B,C,D}. (Sometimes in the code, WC is used for
P and WO for Q.) They have a number of properties:
• P
≥ 0, with P > 0 if and only if [A,B] is controllable, Q ≥ 0 with Q > 0
if and only if [A,C] is observable.
• and
• With vec P denoting the column vector formed by stacking column 1
of P on column 2 on column 3, and so on, and
⊗ denoting Kronecker
product
• The controllability grammian can be thought of as measuring the
difficulty of controlling a system. More specifically, if the system is in
the zero state initially, the minimum energy (as measured by the L
2
norm of u) required to bring it to the state x
0
is x
0
P –1 x
0
; so small
eigenvalues of P correspond to systems that are difficult to control,
while zero eigenvalues correspond to uncontrollable systems.
1 x x – 0 ≤ log +
f t ( ) L
2
∈
f t ( )
2
dt
∫
t
2 ∫
f t ( ) 2
dt
1 2 ⁄
ω 2 ∫
F j ω ( ) 2
d ω
1 2 ⁄
------------------------------------------------------------------------------------------- 4 π ≤
P e
At
BB ′e
A ′t
dt
0
∞
∫
=
Q e
A ′t
C ′Ce
At
dt
0
∞
∫
=
I A A I ⊗
+
⊗ [ ]vecP vec( – BB ′ )
=