National Instruments NI MATRIXx Xmath - Manual (Page 17)

Chapter 1 Introduction
Xmath Model Reduction Module 1-10 ni.com
Internally Balanced Realizations
Suppose that a realization of a transfer-function matrix has the
controllability and observability grammian property that P = Q =
Σ for
some diagonal
Σ. Then the realization is termed internally balanced. Notice
that the diagonal entries
σ
i
of
Σ are square roots of the eigenvalues of PQ,
that is, they are the Hankel singular values. Often the entries of
Σ are
assumed ordered with
σ
i
≥ σ
i+1
.
As noted in the discussion of grammians, systems with small (eigenvalues
of) P are hard to control and those with small (eigenvalues of) Q are hard
to observe. Now a state transformation T =
α I will cause P = Q to be
replaced by
α 2 P,
α –2 Q, implying that ease of control can be obtained at the
expense of difficulty of observation, and conversely. Balanced realizations
are those when ease of control has been balanced against ease of
observation.
Given an arbitrary realization, there are a number of ways of finding a
state-variable coordinate transformation bringing it to balanced form.
A good survey of the available algorithms for balancing is in [LHPW87].
One of these is implemented in the Xmath function
balance( ) .
The one implemented in balmoore( ) as part of this module is more
sophisticated, but more time consuming. It proceeds as follows:
1. Singular value decompositions of P and Q are defined. Because P and
Q are symmetric, this is equivalent to diagonalizing P and Q by
orthogonal transformations.
P = U
c
S
c
U
′
c
Q = U
o
S
o
U
′
o
2. The matrix,
is constructed, and from it, a singular value decomposition is obtained:
3. The balancing transformation is given by:
The balanced realization is T –1 AT, T –1 B, CT.
H S
0
1 2 ⁄
U
H
S
H
V
H
1 2 ⁄
=
H U
H
S
H
V
H
′
=
T U
0
S
0
1 2 ⁄ – U
H
S
H
1 2 ⁄
=