National Instruments NI MATRIXx Xmath - Manual (Page 93)

Chapter 5 Utilities
Xmath Model Reduction Module 5-2 ni.com
The gramian matrices are defined by solving the equations (in continuous
time)
and, in discrete time
The computations are effected with lyapunov( ) and stability is checked,
which is time-consuming. The Hankel singular values are the square roots
of the eigenvalues of the product.
Related Functions
lyapunov() , dlyapunov()
stable( )
[SysS,SysU] = stable(Sys,{tol})
The stable( ) function decomposes Sys into its stable ( SysS ) and
unstable( SysU ) parts, such that Sys=SysS+SysU .
Continuous systems have unstable poles if real parts > –tol .
Discrete systems have unstable poles if magnitudes > 1-tol .
• The direct term (D matrix) is included in SysS .
• If Sys has poles clustered near -tol (or 1-tol ), then SysS and SysU
might be ill-conditioned. To avoid this problem choose tol to a value
that is not close to the majority of poles.
Algorithm
The algorithm begins by transforming the A matrix to Schur form, and
counting the number of stable and unstable eigenvalues, together with
those for which classification is doubtful. Stable eigenvalues are those
in either of the following:
• Re[s] < 0 (continuous time)
• |z| < 1 (discrete time)
AW
c
W
c
A ′
+ BB ′
– =
W
o
A A ′W
0
+ C ′C
– =
W
c
AW
c
A ′
– BB ′
=
W
o
A ′W
o
A
–
C ′C
=