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Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-8 ni.com
Further, the which is optimal for Hankel norm approximation also is
optimal for this second type of approximation.
In Xmath Hankel norm approximation is achieved with
ophank( ).
The most comprehensive reference is [Glo84].
balmoore( )
[SysR,HSV,T] = balmoore(Sys,{nsr,bound})
The balmoore( ) function computes an internally-balanced realization of
a continuous system and then optionally truncates it to provide a balance
reduced order system using B.C. Moore’s algorithm.
When
balmoore( ) is being used to reduce a system, its objective mirrors
that of
redschur( ), therefore, if the same Sys and nsr are used for both
algorithms, the reduced order system should have the same transfer
function (though in general the state-variable realizations will be different).
When
balmoore( ) is being used to balance a system, its objective, like
that of balance, is to generate an internally-balanced state-variable
realization. The implementations are not identical.
balmoore( ) only can be applied on systems that have a stable A matrix,
and are controllable and observable, (that is, minimal). Checks, which are
rather time-consuming, are included. The computation is intrinsically not
well-conditioned if
Sys is nearly nonminimal. The first part of
balmoore( ) serves to find a transformation matrix T such that the
controllability and observability grammians after transformation are equal,
and diagonal, with decreasing entries down the diagonal, that is, the system
representation is internally balanced. (The condition number of T is a
measure of the ill-conditioning of the algorithm. If there is a problem with
ill-conditioning,
redschur( ) can be used as an alternative.) If this
common grammian is Σ, then after transformation:
(continuous)
Σ
A
′
+ A
Σ
= –BB
′Σ
A + A
′Σ
= –C
′
C
(discrete)
Σ
– A
Σ
A
= –BB
′Σ
- A
′ Σ
A = –C
′
C
with with the the Hankel
Singular Values of
Sys. In the second part of balmoore( ), a truncation
G
ˆ
Σ diag σ
1
σ
2
σ
3
... σ
ns
,,,[]= σ
i
σ
i 1+
0>≤σ
i