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<H2><A Name="AB09CX">AB09CX</A></H2>
<H3>
Optimal Hankel-norm approximation based 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,Dr) for a stable
original state-space representation (A,B,C,D) by using the optimal
Hankel-norm approximation method in conjunction with square-root
balancing. The state dynamics matrix A of the original system is
an upper quasi-triangular matrix in real Schur canonical form.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB09CX( DICO, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
$ C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
$ DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
$ M, N, NR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), HSV(*)
</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.
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 TOL1.
</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(MAX(0,NR-KR+1),NMIN),
where KR is the multiplicity of the Hankel singular value
HSV(NR+1), 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(TOL1,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 in a real Schur form.
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).
D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.
LDD INTEGER
The leading dimension of array D. LDD >= 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.
</PRE>
<B>Tolerances</B>
<PRE>
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = 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 TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M), if DICO = 'C';
LIWORK = MAX(1,N,M), if DICO = 'D'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.
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( LDW1,LDW2 ), where
LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2,
LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
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;
= 3: the computation of stable projection failed;
= 4: the order of computed stable projection differs
from the order of Hankel-norm approximation.
</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) + Du(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 AB09CX determines for
the given system (1), the matrices of a reduced order system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(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,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
infinity-norm of G.
The optimal Hankel-norm approximation method of [1], based on the
square-root balancing projection formulas of [2], is employed.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
[2] 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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented methods rely on an accuracy enhancing square-root
technique.
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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