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Chapter 2 Robustness Analysis
MATRIXx Xmath Robust Control Module 2-16 ni.com
VOPT=ssv(M,{scaling="OPT"})
VOPT (a scalar) = 2.43952
VSVD = max(svd(M))
VSVD (a scalar) = 2.65886
osscale( )
[v, vD] = osscale(M)
The osscale( ) function scales a matrix using the Osborne Algorithm.
A diagonal scaling D
OS
is found that minimizes the Frobenius norm of
, which is the square root of the sum of the squares of its
singular values. If M is reducible,
osscale( ) may encounter a divide
by zero. To avoid this, use
ssv( ) with the Osborne scaling option:
[v,vD]=ssv(M,{scaling="OS"})
pfscale( )
[v, vD] = pfscale(M)
The pfscale( ) function scales a matrix using the Perron-Frobenius
Algorithm. This scaling is optimal for matrices with all positive entries.
The matrix M must be irreducible for this function. If M is reducible,
use
ssv( ) with the Perron-Frobenius scaling option instead:
[v,vD]=ssv(M,{scaling="PF"})
The optimum diagonal scaling is found for M using the Perron-Frobenius
theory of non-negative matrices. This scaling is given by
where p and q are right and left eigenvectors of | associated with its largest
eigenvalue:
D
OS
MD
OS
1–
D
i
PF
p
i
q
i
----=
Mp λ
max
p,= M
T
q λ
max
q,= p 0 q≠≠