National Instruments 370755B-01

Chapter 5 Utilities

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 AS and AU are respectively stable and unstable.

A matrix X satisfying –ASX +XAU+ASU = 0 is then determined by calling

the algorithm sylvester( ). The eigenvalue properties of AS and AU

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:

1tol–1tol+,

AASASU

0AU

AIX

→AIX–

AS0

0AU