SSModelReduction( Amat, Bmat, Cmat, Dmat, opts )

Specifies whether to perform the balanced grammian transformation on the input matrices before reduction. If true and the cutoff option is assigned, then the transformation is performed, otherwise it is not.  The default is false. The balanced grammian transformation is a similarity transformation on the A, B, and C matrices that equates their controllability and observability grammians; see DynamicSystems[Grammians] for details.

Specifies the cutoff value for the Hankel singular values in the balanced grammian of the system.  States with associated Hankel singular value smaller than or equal to the cutoff value are eliminated.

Specifies whether the system is discrete or continuous. If true, the system is discrete, otherwise continuous. The default value is false.

Specifies the reduction method. The default is matchDC.

Specifies the states to remove. The default is NULL, which removes no states.

If removestate = $m$, with $m$ a positive integer, then the model states from $m$ to $n$ are removed. The order of the reduced system is $r=m-1$.

If removestate is a list of positive integers, then it represents the list of indexes corresponding to the states to be eliminated.

If removestate is a list of truefalse values and its length is the same as the length ($n$) of the state vector $X\left(t\right)$, then each true in the list removes the corresponding state in $X\left(t\right)$.

The removestate option is ignored if both the removestate and the cutoff options are specified. If both the removestate and the cutoff options are omitted, no reduction is performed and the input matrices are returned without modification.

$\mathrm{Ar}=\mathrm{A11}$

$\mathrm{Br}=\mathrm{B1}$

$\mathrm{Cr}=\mathrm{C1}$

$\mathrm{Dr}=\mathrm{D}$

where the submatrices $\mathrm{A11}$, $\mathrm{B1}$, and $\mathrm{C1}$ are defined by the following partitioning of the original state-space matrices:

$A=\mathrm{Matrix}\left(\left[\left[\mathrm{A11}\,\mathrm{A12}\right]\,\left[\mathrm{A21}\,\mathrm{A22}\right]\right]\right)$

$B=\mathrm{Matrix}\left(\left[\left[\mathrm{B1}\right]\,\left[\mathrm{B2}\right]\right]\right)$

$C=\mathrm{Matrix}\left(\left[\left[\mathrm{C1}\,\mathrm{C2}\right]\right]\right)$

where $\mathrm{A11}$ has dimension $r$ x $r$, $\mathrm{B1}$ has dimension $r$ x $i$, and $\mathrm{C1}$ has dimension $o$ x $r$.

Ar = ( A11 + signK * A12 . A22inv . A21 )

Br = ( B1 + signK * A12 . A22inv . B2 )

Cr = ( C1 + signK * C2 . A22inv . A21 )

Dr = ( D + signK * C2 . A22inv . B2 )

where for continuous systems signK = -1 and A22inv = A22^(-1) while for discrete systems signK = +1 and A22inv = (I - A22)^(-1), with $\mathrm{I}$ the identity matrix of appropriate dimension. The matchDC method fails if $\mathrm{A22}$ (for continuous systems) or $\mathrm{I}-\mathrm{A22}$ (for discrete systems) is singular.