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<H2><A Name="AB09AX">AB09AX</A></H2>
<H3>
Balance & Truncate model reduction for stable systems with state matrix in real Schur canonical form
</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 a reduced order model (Ar,Br,Cr) for a stable original
state-space representation (A,B,C) by using either the square-root
or the balancing-free square-root Balance & Truncate model
reduction method. The state dynamics matrix A of the original
system is an upper quasi-triangular matrix in real Schur canonical
form. The matrices of the reduced order system are computed using
the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T .
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, HSV, T, LDT, TI, LDTI, TOL, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOB, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDT, LDTI, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*),
$ T(LDT,*), TI(LDTI,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
DICO CHARACTER*1
Specifies the type of the original system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOB CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root Balance & Truncate method;
= 'N': use the balancing-free square-root
Balance & Truncate method.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting order NR is fixed;
= 'A': the resulting order NR is automatically determined
on basis of the given tolerance TOL.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A. N >= 0.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NR (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system. 0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).
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 a real Schur
canonical form.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the
reduced order system.
LDA INTEGER
The leading dimension of 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 original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.
LDB INTEGER
The leading dimension of 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 original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.
T (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T.
LDT INTEGER
The leading dimension of array T. LDT >= MAX(1,N).
TI (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI.
LDTI INTEGER
The leading dimension of array TI. LDTI >= MAX(1,N).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL <= 0 on entry.
If ORDSEL = 'F', the value of TOL is ignored.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B', or
LIWORK = N, if JOB = 'N'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX(1,N*(MAX(N,M,P)+5) + N*(N+1)/2).
For optimum performance LDWORK should be larger.
</PRE>
<B>Warning Indicator</B>
<PRE>
IWARN INTEGER
= 0: no warning;
= 1: with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.
</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 state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2: the computation of Hankel singular values failed.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Let be the stable linear system
d[x(t)] = Ax(t) + Bu(t)
y(t) = Cx(t) (1)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09AX determines for
the given system (1), the matrices of a reduced NR order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) (2)
such that
HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.
If JOB = 'B', the square-root Balance & Truncate method of [1]
is used and, for DICO = 'C', the resulting model is balanced.
By setting TOL <= 0, the routine can be used to compute balanced
minimal state-space realizations of stable systems.
If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method [2] is used.
By setting TOL <= 0, the routine can be used to compute minimal
state-space realizations of stable systems.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
[2] Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N 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>
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