<HTML>
<HEAD><TITLE>TB01KX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="TB01KX">TB01KX</A></H2>
<H3>
Additive spectral decomposition of the transfer-function matrix of a standard system
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute an additive spectral decomposition of the transfer-
function matrix of the system (A,B,C) by reducing the system
state-matrix A to a block-diagonal form. It is assumed that A is
in a real Schur form, and the leading diagonal block of order NDIM
has eigenvalues distinct from those of the trailing diagonal
block. The system matrices are transformed as
A <-- V*A*U, B <--V*B and C <-- C*U, where V = inv(U),
preserving the spectra of the two diagonal blocks.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE TB01KX( N, M, P, NDIM, A, LDA, B, LDB, C, LDC, U, LDU,
$ V, LDV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDU, LDV, M, N, NDIM, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), U(LDU,*), V(LDV,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the state-space representation, i.e., the
order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs, or of columns of B. M >= 0.
P (input) INTEGER
The number of system outputs, or of rows of C. P >= 0.
NDIM (input) INTEGER
The dimension of the leading diagonal block of A having
eigenvalues distinct from those of the trailing diagonal
block. 0 <= NDIM <= N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A in real Schur form.
On exit, the leading N-by-N part of this array contains a
block diagonal matrix inv(U) * A * U with two diagonal
blocks in real Schur form, with the elements below the
first subdiagonal set to zero. The leading block has
dimension NDIM-by-NDIM.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix inv(U) * B.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * U.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain an initial transformation matrix U.
On exit, the leading N-by-N part of this array contains
the transformation matrix used to reduce A to the block-
diagonal form. The first NDIM columns of U span the
invariant subspace of A corresponding to the eigenvalues
of its leading diagonal block. The last N-NDIM columns of
U span the reducing subspace of A corresponding to the
eigenvalues of the trailing diagonal block of A.
LDU INTEGER
The leading dimension of the array U. LDU >= max(1,N).
V (output) DOUBLE PRECISION array, dimension (LDV,N)
The leading N-by-N part of this array contains the
inverse of the transformation matrix U used to reduce A
to the block-diagonal form.
LDV INTEGER
The leading dimension of the array V. LDV >= max(1,N).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the separation of the two diagonal blocks failed
because of very close eigenvalues.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
A similarity transformation U is determined that reduces the given
system state-matrix A to a block-diagonal form (with two diagonal
blocks), so that the eigenvalues of the leading diagonal block of
the resulting A are preserved. The determined transformation is
applied to the system (A,B,C) as
A <-- inv(U)*A*U, B <-- inv(U)*B and C <-- C*U.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Safonov, M.G., Jonckheere, E.A., Verma, M., Limebeer, D.J.N.
Synthesis of positive real multivariable feedback systems.
Int. J. Control, pp. 817-842, 1987.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm requires about N /2 + NDIM*(N-NDIM)*(2*N+M+P)
floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
<A HREF=support.html><B>Return to Supporting Routines index</B></A></BODY>
</HTML>