National Instruments 370757C-01 Manual De Usuario
Chapter 2
Robustness Analysis
© National Instruments Corporation
2-15
ssv( )
[v,vD] = SSV(M, {scaling})
The
ssv( )
function computes an approximation (and guaranteed upper
bound) to the Scaled Singular Value of a complex square matrix M, where
M can be a reducible matrix. The scaled singular value v(M) is defined by:
M can be a reducible matrix. The scaled singular value v(M) is defined by:
Scaling can be accomplished with one of three algorithms:
•
Perron-Frobenius—If
{scaling="PF"}
Safonov’s
Perron-Frobenius method [Saf82] is used. This method finds the scaled
singular value for non-negative real matrices M. In general, it is
suboptimal if M is complex. This algorithm is the default because
empirical tests show that is the fastest of the three.
singular value for non-negative real matrices M. In general, it is
suboptimal if M is complex. This algorithm is the default because
empirical tests show that is the fastest of the three.
•
Osborne—If
{scaling="OS"}
, Osborne’s Method [Osb60] is used.
This method solves the problem of finding D
O
such that
where D is diagonal and positive, and
is the Frobenius norm.
Thus, the Osborne method minimizes the Frobenius norm, and is
therefore suboptimal.
therefore suboptimal.
•
Optimal—If
{scaling="OPT"}
, Boyd’s ellipsoid algorithm
[BYB89] is used. This algorithm computes the scaled singular value
to a guaranteed accuracy. It is, however, the most computationally
expensive of the three algorithms.
to a guaranteed accuracy. It is, however, the most computationally
expensive of the three algorithms.
ssv( ) Examples
Consider the complex matrix M:
M = [–1, jay, 0; 0, 2*jay, 1+jay;1, 0, 1];
ssv( )
can return the optimally scaled singular value of M using Osborne,
Perron-Frobenius, or Boyd methods:
VOS=ssv(M,{scaling="OS"})
VOS (a scalar) = 2.56723
VPF=ssv(M,{scaling="PF"})
VPF (a scalar) = 2.45133
v M
( ) =
inf
σ DMD
1
–
(
)
D
C
n n
×
∈
det D
( ) 0 dia
,
≠
,
gonal
D
O
MD
O
1
–
inf
D diagonal
DMD
1
–
F
⋅
F