National Instruments NI MATRIXx Xmath User Manual
Page 94
Chapter 5
Utilities
© National Instruments Corporation
5-3
Doubtful ones are those for which the real part of the eigenvalue has
magnitude less than or equal to
tol
for continuous-time, or eigenvalue
magnitude within the following range for discrete time:
A warning is given if doubtful eigenvalues exist.
The algorithm then computes a real ordered Schur decomposition of A
so that after transformation
where the eigenvalues of A
S
and A
U
are respectively stable and unstable.
A matrix X satisfying –A
SX
+ XA
U
+ A
SU
= 0 is then determined by calling
the algorithm
sylvester( )
. The eigenvalue properties of A
S
and A
U
guarantee that X exists. If doubtful eigenvalues are present, they are
assigned to the unstable part of
Sys
. In this circumstance you get the
message,
The system has poles near, or upon, the jw-axis
for continuous systems, and the following for discrete systems:
The system has poles near the unit circle.
Note
If A has eigenvalues clustered near
-tol
(
1–tol
in discrete-time), then X is likely
to be ill-conditioned and consequently
SysS
and
SysU
will also be ill-conditioned. (For
example, the B matrix of
SysS
could contain very small values, while the C matrix could
contain large values. In this case,
SysS
would be very weakly controllable and very
strongly observable. This will cause problems when gramians and Hankel singular values
are calculated.) To avoid this problem, change
tol
to a value that is not close to the
majority of eigenvalues.
A further transformation of A is constructed using X:
1 tol
–
1 tol
+
,
A
A
S
A
SU
0 A
U
=
A
I X
0 I
→
A I X
–
0 I
A
S
0
0 A
U
=