National Instruments NI MATRIXx Xmath User Manual
Page 52
Chapter 3
Multiplicative Error Reduction
3-6
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2.
With G(s) = D + C(sI – A)
–1
B and stable, with DD´ nonsingular and
G(j
ω) G'(–jω) nonsingular for all ω, part of a state variable realization
of a minimum phase stable W(s) is determined such that
W´(–s)W(s) = G(s)G´(–s) with
The state variable matrices in W(s) are obtained as follows. The
controllability grammian P associated with G(s) is first found from
AP + PA´ + BB´=0 then A
W
= A, B
W
= PC´ + BD´.
When G(s) is square, the algorithm checks to see if there is a zero or
singularity of G(s) close to the j
ω-axis (the zeros are given by the
eigenvalues of A – BD
–1
C and are computed reliably with the aid of
schur( )
). If one is found, you are warned that results may be
unreliable. Next, a stabilizing solution Q is found for the following
Riccati equation:
The
singriccati( )
function is used; failure of the nonsingularity
condition on G(j
ω)G´(–jω) will normally result in an error message
that no stabilizing solution exists. To obtain the best numerical results,
singriccati( )
is invoked with the keyword
{method="schur"}
.
Although D
W
, C
W
are not needed for the remainder of the algorithm,
they are simply determined in the square case by
with minor modification in the nonsquare case. The real point of the
algorithm is to compute P and Q; the matrix Q satisfies (square or
nonsquare case).
P, Q are the controllability and observability grammians of the transfer
function C
W
(sI – A)
–1
B. This transfer function matrix, it turns out, is
the strictly proper, stable part of
θ(s) = W
–T
(–s)G(s), which obeys the
matrix all-pass property
θ(s)θ´(–s) = I, and is the phase matrix
associated with G(s).
3.
Compute ordered Schur decompositions of PQ, with the eigenvalues
of PQ is ascending and descending order. Obtain the phase matrix
Hankel singular values, that is, the Hankel singular values of the
W s
( )
D
W
C
W
sI A
w
–
(
)
1
–
B
W
+
=
QA A
′
Q
C B
′
WQ
–
(
)′ DD′
(
)
1
–
C B
W
′
Q
–
(
)
+
+
0
=
D
W
D
′
C
W
D
1
–
C B
W
′
Q
–
(
)
=
=
QA A
′
Q C
W
′
C
W
+
+
0
=