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DynamicSystems[SSModelReduction] - reduce a state-space system

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

Parameters

 Amat - Matrix; system matrix of a state-space system Bmat - Matrix; input matrix of a state-space system Cmat - Matrix; output matrix of a state-space system Dmat - (optional) Matrix; direct-through matrix of a state-space system. If not specified, a zero-matrix of appropriate dimension is used. opts - (optional) equation(s) of the form option = value; specify options for the SSModelReduction command

Description

 • The SSModelReduction command returns reduced-order state-space system matrices from a given set of state-space system matrices, Amat, Bmat, Cmat, and Dmat.
 • For a state-space system defined by the state and output equations $\frac{ⅆ}{ⅆt}X\left(t\right)=AX\left(t\right)+BU\left(t\right)$ and $Y\left(t\right)=CX\left(t\right)+\mathrm{D}U\left(t\right)$, respectively, the state vector X(t) with dimension n can be partitioned into $X\left(t\right)=\mathrm{Vector}\left(\left[\mathrm{X1},\mathrm{X2}\right]\right)$, where $\mathrm{X1}$ contains the states to be retained and   $\mathrm{X2}$ contains the states to be removed. The dimension of $\mathrm{X1}$, referred to here as $r$, is the reduced state order. The number of input of the system is denoted as $i$, while the number of output is denoted as $o$.
 • Two options, cutoff and removestate determine the number of states to be removed. Only one is used; if both are specified, cutoff is used. See the Options section for a description of their operation.
 • Two reduction methods are supported:  truncate and matchDC.
 • The truncate method returns the reduced state-space matrices by simply removing the selected states. The reduced state-space matrices are given by:

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

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

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

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

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 $r$.

 • The matchDC method computes the reduced state-space matrices such that the DC gain (steady-state response) of the reduced-order system matches that of the original system. With the same partition as described for the truncate option, the matchDC reduced state-space matrices are

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 $I$ the identity matrix of appropriate dimension. The matchDC method fails if $\mathrm{A22}$ (for continuous systems) or $I-\mathrm{A22}$ (for discrete systems) is singular.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{Amat}:=⟨⟨-5|1|0⟩,⟨0|-2|1⟩,⟨0|0|-1⟩⟩:$
 > $\mathrm{Bmat}:=⟨⟨0,0,1⟩⟩:$
 > $\mathrm{Cmat}:=⟨⟨1|0|0⟩⟩:$
 > $\mathrm{Dmat}:=⟨⟨0⟩⟩:$
 > $\mathrm{SSModelReduction}\left(\mathrm{Amat},\mathrm{Bmat},\mathrm{Cmat},\mathrm{Dmat},'\mathrm{removestate}'=2,\mathrm{method}=\mathrm{matchDC}\right)$
 $\left[\begin{array}{r}{-}{5}\end{array}\right]{,}\left[\begin{array}{c}\frac{{1}}{{2}}\end{array}\right]{,}\left[\begin{array}{r}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\end{array}\right]$ (1)
 > $\mathrm{SSModelReduction}\left(\mathrm{Amat},\mathrm{Bmat},\mathrm{Cmat},\mathrm{Dmat},'\mathrm{removestate}'=2,\mathrm{method}=\mathrm{truncate}\right)$
 $\left[\begin{array}{r}{-}{5}\end{array}\right]{,}\left[\begin{array}{r}{0}\end{array}\right]{,}\left[\begin{array}{r}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\end{array}\right]$ (2)