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<HTML>
<HEAD><TITLE>AB13HD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB13HD">AB13HD</A></H2>
<H3>
L-infinity norm of a state space system in standard or in descriptor 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 the L-infinity norm of a proper continuous-time or
causal discrete-time system, either standard or in the descriptor
form,
-1
G(lambda) = C*( lambda*E - A ) *B + D .
The norm is finite if and only if the matrix pair (A,E) has no
finite eigenvalue on the boundary of the stability domain, i.e.,
the imaginary axis, or the unit circle, respectively.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB13HD( DICO, JOBE, EQUIL, JOBD, CKPROP, REDUCE, POLES,
$ N, M, P, RANKE, FPEAK, A, LDA, E, LDE, B, LDB,
$ C, LDC, D, LDD, NR, GPEAK, TOL, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER CKPROP, DICO, EQUIL, JOBD, JOBE, POLES, REDUCE
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ LZWORK, M, N, NR, P, RANKE
C .. Array Arguments ..
COMPLEX*16 ZWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), E( LDE, * ),
$ FPEAK( 2 ), GPEAK( 2 ), TOL( * )
INTEGER IWORK( * )
LOGICAL BWORK( * )
</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 system, as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBE CHARACTER*1
Specifies whether E is an identity matrix, a general
square matrix, or a matrix in compressed form, as follows:
= 'I': E is the identity matrix;
= 'G': E is a general matrix;
= 'C': E is in compressed form, i.e., E = [ T 0 ],
[ 0 0 ]
with a square full-rank matrix T.
EQUIL CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the system (A,E,B,C) or (A,B,C), as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears in
the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix;
= 'F': D is known to be well-conditioned (hence, to have
full rank), for DICO = 'C' and JOBE = 'I'.
The options JOBD = 'D' and JOBD = 'F' produce the same
results, but much less memory is needed for JOBD = 'F'.
CKPROP CHARACTER*1
If DICO = 'C' and JOBE <> 'I', specifies whether the user
wishes to check the properness of the transfer function of
the descriptor system, as follows:
= 'C': check the properness;
= 'N': do not check the properness.
If the test is requested and the system is found improper
then GPEAK and FPEAK are both set to infinity, i.e., their
second component is zero; in addition, IWARN is set to 2.
If the test is not requested, but the system is improper,
the resulted GPEAK and FPEAK may be wrong.
If DICO = 'D' or JOBE = 'I', this option is ineffective.
REDUCE CHARACTER*1
If CKPROP = 'C', specifies whether the user wishes to
reduce the system order, by removing all uncontrollable
and unobservable poles before computing the norm, as
follows:
= 'R': reduce the system order;
= 'N': compute the norm without reducing the order.
If CKPROP = 'N', this option is ineffective.
POLES CHARACTER*1
Specifies whether the user wishes to use all or part of
the poles to compute the test frequencies (in the non-
iterative part of the algorithm), or all or part of the
midpoints (in the iterative part of the algorithm), as
follows:
= 'A': use all poles with non-negative imaginary parts
and all midpoints;
= 'P': use part of the poles and midpoints.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
P (input) INTEGER
The row size of the matrix C. P >= 0.
RANKE (input) INTEGER
If JOBE = 'C', RANKE denotes the rank of the descriptor
matrix E or the size of the full-rank block T.
0 <= RANKE <= N.
FPEAK (input/output) DOUBLE PRECISION array, dimension (2)
On entry, this parameter must contain an estimate of the
frequency where the gain of the frequency response would
achieve its peak value. Setting FPEAK(2) = 0 indicates an
infinite frequency. An accurate estimate could reduce the
number of iterations of the iterative algorithm. If no
estimate is available, set FPEAK(1) = 0, and FPEAK(2) = 1.
FPEAK(1) >= 0, FPEAK(2) >= 0.
On exit, if INFO = 0, this array contains the frequency
OMEGA, where the gain of the frequency response achieves
its peak value GPEAK, i.e.,
|| G ( j*OMEGA ) || = GPEAK , if DICO = 'C', or
j*OMEGA
|| G ( e ) || = GPEAK , if DICO = 'D',
where OMEGA = FPEAK(1), if FPEAK(2) > 0, and OMEGA is
infinite, if FPEAK(2) = 0. (If nonzero, FPEAK(2) = 1.)
For discrete-time systems, it is assumed that the sampling
period is Ts = 1. If Ts <> 1, the frequency corresponding
to the peak gain is OMEGA/Ts.
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.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
N-by-N part of this array contains the state dynamics
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading NR-by-NR part of this array contains the state
dynamics matrix of an equivalent reduced, possibly scaled
(if EQUIL = 'S') system, used to check the properness.
Otherwise, the array A is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E (input/output) DOUBLE PRECISION array, dimension (LDE,K),
where K is N, RANKE, or 0, if JOBE = 'G', 'C', or 'I',
respectively.
On entry, if JOBE = 'G', the leading N-by-N part of this
array must contain the descriptor matrix E of the system.
If JOBE = 'C', the leading RANKE-by-RANKE part of this
array must contain the full-rank block T of the descriptor
matrix E.
If JOBE = 'I', then E is assumed to be the identity matrix
and is not referenced.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
K-by-K part of this array contains the descriptor matrix
of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading MIN(K,NR)-by-MIN(K,NR) part of this array contains
the descriptor matrix of an equivalent reduced, possibly
scaled (if EQUIL = 'S') system, used to check the
properness.
Otherwise, the array E is unchanged.
LDE INTEGER
The leading dimension of the array E.
LDE >= MAX(1,N), if JOBE = 'G';
LDE >= MAX(1,RANKE), if JOBE = 'C';
LDE >= 1, if JOBE = 'I'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the system input matrix B.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
NR-by-M part of this array contains the system input
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading NR-by-M part of this array contains the system
input matrix of an equivalent reduced, possibly scaled (if
EQUIL = 'S') system, used to check the properness.
Otherwise, the array B is unchanged.
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 system output matrix C.
On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
P-by-NR part of this array contains the system output
matrix of an equivalent, scaled system.
On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
leading P-by-NR part of this array contains the system
output matrix of an equivalent reduced, possibly scaled
(if EQUIL = 'S') system, used to check the properness.
Otherwise, the array C is unchanged.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D' or JOBD = 'F', the leading P-by-M part of
this array must contain the direct transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D' or JOBD = 'F';
LDD >= 1, if JOBD = 'Z'.
NR (output) INTEGER
If CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the order of
the reduced system. Otherwise, NR = N.
GPEAK (output) DOUBLE PRECISION array, dimension (2)
The L-infinity norm of the system, i.e., the peak gain
of the frequency response (as measured by the largest
singular value in the MIMO case), coded in the same way
as FPEAK.
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION array, dimension K, where K = 2, if
CKPROP = 'N' or DICO = 'D' or JOBE = 'I', and K = 4,
otherwise.
TOL(1) is the tolerance used to set the accuracy in
determining the norm. 0 <= TOL(1) < 1.
TOL(2) is the threshold value for magnitude of the matrix
elements, if EQUIL = 'S': elements with magnitude less
than or equal to TOL(2) are ignored for scaling. If the
user sets TOL(2) >= 0, then the given value of TOL(2) is
used. If the user sets TOL(2) < 0, then an implicitly
computed, default threshold, THRESH, is used instead,
defined by THRESH = 0.1, if MN/MX < EPS, and otherwise,
THRESH = MIN( 100*(MN/(EPS**0.25*MX))**0.5, 0.1 ), where
MX and MN are the maximum and the minimum nonzero absolute
value, respectively, of the elements of A and E, and EPS
is the machine precision (see LAPACK Library routine
DLAMCH). TOL(2) = 0 is not always a good choice.
TOL(2) < 1. TOL(2) is not used if EQUIL = 'N'.
TOL(3) is the tolerance to be used in rank determinations
when transforming (lambda*E-A,B,C), if CKPROP = 'C'. If
the user sets TOL(3) > 0, then the given value of TOL(3)
is used as a lower bound for reciprocal condition numbers
in rank determinations; a (sub)matrix whose estimated
condition number is less than 1/TOL(3) is considered to be
of full rank. If the user sets TOL(3) <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF1 = N*N*EPS, is used instead. TOL(3) < 1.
TOL(4) is the tolerance to be used for checking the
singularity of the matrices A and E when CKPROP = 'C'.
If the user sets TOL(4) > 0, then the given value of
TOL(4) is used. If the user sets TOL(4) <= 0, then an
implicitly computed, default tolerance, defined by
TOLDEF2 = N*EPS, is used instead. The 1-norms of A and E
are also taken into account. TOL(4) < 1.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
LIWORK >= 1, if MIN(N,P,M) = 0, or B = 0, or C = 0; else
LIWORK >= MAX(1,N), if DICO = 'C', JOBE = 'I', and
JOBD <> 'D';
LIWORK >= 2*N + M + P + R + 12, otherwise, where
R = 0, if M + P is even,
R = 1, if M + P is odd.
On exit, if INFO = 0, IWORK(1) returns the number of
iterations performed by the iterative algorithm
(possibly 0).
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
On exit, if INFO = -28, DWORK(1) returns the minimum
value of LDWORK. These values are also set when LDWORK = 0
on entry, but no error message related to LDWORK is issued
by XERBLA.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if MIN(M,P) = 0 or ( JOBD = 'Z' and
( N = 0 or B = 0 or C = 0 ) );
LDWORK >= P*M + x, if ( ( N = 0 and MIN(M,P) > 0 )
or ( B = 0 or C = 0 ) ) and JOBD <> 'Z',
where
x = MAX( 4*MIN(M,P) + MAX(M,P), 6*MIN(M,P) ),
if DICO = 'C',
x = 6*MIN(M,P), if DICO = 'D';
LDWORK >= MAX( 1, N*(N+M+P+2) + MAX( N*(N+M+2) + P*M + x,
4*N*N + 9*N ) ),
if DICO = 'C', JOBE = 'I' and JOBD = 'Z'.
LDWORK >= MAX( 1, (N+M)*(M+P) + P*P + x,
2*N*(N+M+P+1) + N + MIN(P,M) +
MAX( M*(N+P) + N + x, N*N +
MAX( N*(P+M) + MAX(M,P),
2*N*N + 8*N ) ) ),
if DICO = 'C', JOBE = 'I' and JOBD = 'F'.
The formulas for other cases, e.g., for JOBE <> 'I' or
CKPROP = 'C', contain additional and/or other terms.
The minimum value of LDWORK for all cases can be obtained
in DWORK(1) when LDWORK is set to 0 on entry.
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) contains the optimal
LZWORK.
On exit, if INFO = -30, ZWORK(1) returns the minimum
value of LZWORK. These values are also set when LZWORK = 0
on entry, but no error message related to LZWORK is issued
by XERBLA.
If LDWORK = 0 and LZWORK = 0 are both set on entry, then
on exit, INFO = -30, but both DWORK(1) and ZWORK(1) are
set the minimum values of LDWORK and LZWORK, respectively.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if MIN(N,M,P) = 0, or B = 0, or C = 0;
LZWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(M,P) + MAX(M,P)),
otherwise.
For good performance, LZWORK must generally be larger.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (N)
</PRE>
<B>Warning Indicator</B>
<PRE>
IWARN INTEGER
= 0: no warning;
= 1: the descriptor system is singular. GPEAK(1) and
GPEAK(2) are set to 0. FPEAK(1) and FPEAK(2) are
set to 0 and 1, respectively;
= 2: the descriptor system is improper. GPEAK(1) and
GPEAK(2) are set to 1 and 0, respectively,
corresponding to infinity. FPEAK(1) and FPEAK(2) are
set similarly. This warning can only appear if
CKPROP = 'C'.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: a matrix is (numerically) singular or the Sylvester
equation is very ill-conditioned, when computing the
largest singular value of G(infinity) (for
DICO = 'C'); the descriptor system is nearly
singular; the L-infinity norm could be infinite;
= 2: the (periodic) QR (or QZ) algorithm for computing
eigenvalues did not converge;
= 3: the SVD algorithm for computing singular values did
not converge;
= 4: the tolerance is too small and the algorithm did
not converge; this is a warning;
= 5: other computations than QZ iteration, or reordering
of eigenvalues, failed in the LAPACK Library
routines DHGEQZ or DTGSEN, respectively;
= 6: the numbers of "finite" eigenvalues before and after
reordering differ; the threshold used might be
unsuitable.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine implements the method presented in [2], which is an
extension of the method in [1] for descriptor systems. There are
several improvements and refinements [3-5] to increase numerical
robustness, accuracy and efficiency, such as the usage of
structure-preserving eigenvalue computations for skew-Hamiltonian/
Hamiltonian eigenvalue problems in the iterative method in [2].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Bruinsma, N.A. and Steinbuch, M.
A fast algorithm to compute the H-infinity-norm of a transfer
function matrix.
Systems & Control Letters, vol. 14, pp. 287-293, 1990.
[2] Voigt, M.
L-infinity-Norm Computation for Descriptor Systems.
Diploma Thesis, Fakultaet fuer Mathematik, TU Chemnitz,
http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001050.
[3] Benner, P., Sima, V. and Voigt, M.
L-infinity-norm computation for continuous-time descriptor
systems using structured matrix pencils.
IEEE Trans. Auto. Contr., AC-57, pp.233-238, 2012.
[4] Benner, P., Sima, V. and Voigt, M.
Robust and efficient algorithms for L-infinity-norm
computations for descriptor systems.
7th IFAC Symposium on Robust Control Design (ROCOND'12),
pp. 189-194, 2012.
[5] Benner, P., Sima, V. and Voigt, M.
Algorithm 961: Fortran 77 subroutines for the solution of
skew-Hamiltonian/Hamiltonian eigenproblems.
ACM Trans. Math. Softw, 42, pp. 1-26, 2016.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
If the algorithm does not converge in MAXIT = 30 iterations
(INFO = 4), the tolerance must be increased, or the system is
improper.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
Setting POLES = 'P' usually saves some computational effort. The
number of poles used is defined by the parameters BM, BNEICD,
BNEICM, BNEICX, BNEIR and SWNEIC.
Both real and complex optimal workspace sizes are computed if
either LDWORK = -1 or LZWORK = -1.
</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>
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