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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. 

lyapunov()

, 

dlyapunov()

[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.

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)

′

+

′

–

=

′W

0

+

′C

–

=

′

–

′

=

′W

–

′C

=