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IEEE
TRANSACTIONS
ON
AUTOMATIC
CONTROL.
VOL. AC32.
NO.
2, FEBRUARY
1987
115
Computation
of
System Balancing Transformations

and
Other Applications
Simultaneous
Diagonalization Algorithms
ALAN J. LAUB,
FELLOW, IEEE,
MICHAEL
T.
HEATH, CHRIS C. PAIGE,
AND
RO3ERT
c.
WARD
AbstractAn
algorithm is presented in this paper for
computing
state
space balancing transformations directly from
a statespace realization.
The algorithm requires no “squaring up” or unnecessary matrix prod
ucts. Various algorithmic aspects are discussed in detail. A key feature of
the algorithm is the determination of
a contragredient transformation
through computing the singular value decomposition
a
certain product
of matrices without explicitly forming the product. Other contragredient
transformation applications are also described. It
is further shown that a
similar approach may be taken. involving the generalized singular value
decomposition,
to
the classical simultaneous diagonalization problem.
These SVDbased simultaneous
diagonalization
algorithms provide a
computational alternative to existing methods for solving certain classes
of symmetric positive definite generalized eigenvalue problems.
I.
IKTRODUCTION
T
HE
notion of balancing in linear systems
and
control
theory
has become
a
cornerstone
for important new theoretical
developments with farreaching
implications.
Some
early
ideas
were
discussed
by Moore in [I91 and further
refinements,
extensions. and applications have appeared in. for
example.
[4],
[6][8],
[13], [17], [22], 1251[27], [29] and the
references
therein. The literature on the
subject is
now voluminous.
One
of the main reasons why balancing is
of interest in control
and
one of its principal
applications is in
model reduction. For
statespace
models,
a
methodology
for
deriving
reducedorder
models is provided in terms of a
system‘s
realization in balanced
coordinates.
The key computational
problem
there is
the calcula
tion of a
balancing
transformation
and
the
matrices
of the balanced
realization. Other
applications
include
a
solution
to the
Hankel
norm approximation
problem and other
frequency
domain
and
frequency respmse approximation
problems.
The key paper by
Glover [8] provides many further details andreferences.Some
important new advances are described in [8] and
much
of
the
previous work is tied together in averyreadable
and coherent
presentation.
Other
applications
of balancing
transformations
(although not
by
that
name)
can be found in signal
processing;
see,
for
example. [20].
In this paper we shall concentrate on describing
a
numerical
algorithm
for
computing
a
statespace
balancing
transformation.
Other
algorithms,
particularly
some
early
ideas
in [I61 which
motivated this paper, and numerous
related
aspects will also be
Manuscript received January
13.
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