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(jω) denotes the Fourier transform of some , the
Heisenberg inequality states,
and the bound is tight since it is attained for f(t) = exp + (–kt2).
Commonly Used Concepts
This section outlines some frequently used standard concepts.

Controllability and Observability Grammians

Suppose that G(s)=D+C(sI–A)–1B is a transfer-function matrix with
Reλi(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 L2
norm of u) required to bring it to the state x0 is x0P –1x0; so small
eigenvalues of P correspond to systems that are difficult to control,
while zero eigenvalues correspond to uncontrollable systems.
1xx–0≤log+
ft() L2
∈
ft()
2dt
∫
t2
∫ft()2dt
12⁄
ω2
∫Fjω()
2dω
12⁄
-------------------------------------------------------------------------------------------4π≤
Pe
AtBB′eA′tdt
0
∞
∫
=Qe
A′tC′CeAtdt
0
∞
∫
=
IAAI⊗+⊗[]vecPvec(–BB′)=