350 lines
12 KiB
Fortran
350 lines
12 KiB
Fortran
DOUBLE PRECISION FUNCTION AB13AD( DICO, EQUIL, N, M, P, ALPHA, A,
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$ LDA, B, LDB, C, LDC, NS, HSV,
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$ DWORK, LDWORK, INFO )
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C
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C SLICOT RELEASE 5.0.
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C
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C Copyright (c) 2002-2009 NICONET e.V.
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C
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C This program is free software: you can redistribute it and/or
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C modify it under the terms of the GNU General Public License as
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C published by the Free Software Foundation, either version 2 of
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C the License, or (at your option) any later version.
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C
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C This program is distributed in the hope that it will be useful,
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C but WITHOUT ANY WARRANTY; without even the implied warranty of
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C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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C GNU General Public License for more details.
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C
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C You should have received a copy of the GNU General Public License
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C along with this program. If not, see
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C <http://www.gnu.org/licenses/>.
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C
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C PURPOSE
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C
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C To compute the Hankel-norm of the ALPHA-stable projection of the
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C transfer-function matrix G of the state-space system (A,B,C).
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C
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C FUNCTION VALUE
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C
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C AB13AD DOUBLE PRECISION
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C The Hankel-norm of the ALPHA-stable projection of G
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C (if INFO = 0).
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C
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C ARGUMENTS
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C
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C Mode Parameters
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C
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C DICO CHARACTER*1
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C Specifies the type of the system as follows:
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C = 'C': continuous-time system;
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C = 'D': discrete-time system.
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C
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C EQUIL CHARACTER*1
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C Specifies whether the user wishes to preliminarily
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C equilibrate the triplet (A,B,C) as follows:
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C = 'S': perform equilibration (scaling);
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C = 'N': do not perform equilibration.
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C
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C Input/Output Parameters
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C
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C N (input) INTEGER
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C The order of the state-space representation, i.e. the
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C order of the matrix A. N >= 0.
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C
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C M (input) INTEGER
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C The number of system inputs. M >= 0.
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C
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C P (input) INTEGER
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C The number of system outputs. P >= 0.
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C
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C ALPHA (input) DOUBLE PRECISION
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C Specifies the ALPHA-stability boundary for the eigenvalues
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C of the state dynamics matrix A. For a continuous-time
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C system (DICO = 'C'), ALPHA <= 0 is the boundary value for
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C the real parts of eigenvalues, while for a discrete-time
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C system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
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C boundary value for the moduli of eigenvalues.
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C The ALPHA-stability domain does not include the boundary
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C (see the Note below).
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C
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C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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C On entry, the leading N-by-N part of this array must
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C contain the state dynamics matrix A.
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C On exit, if INFO = 0, the leading N-by-N part of this
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C array contains the state dynamics matrix A in a block
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C diagonal real Schur form with its eigenvalues reordered
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C and separated. The resulting A has two diagonal blocks.
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C The leading NS-by-NS part of A has eigenvalues in the
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C ALPHA-stability domain and the trailing (N-NS) x (N-NS)
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C part has eigenvalues outside the ALPHA-stability domain.
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C Note: The ALPHA-stability domain is defined either
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C as the open half complex plane left to ALPHA,
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C for a continous-time system (DICO = 'C'), or the
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C interior of the ALPHA-radius circle centered in the
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C origin, for a discrete-time system (DICO = 'D').
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C
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C LDA INTEGER
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C The leading dimension of array A. LDA >= MAX(1,N).
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C
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C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
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C On entry, the leading N-by-M part of this array must
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C contain the original input/state matrix B.
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C On exit, if INFO = 0, the leading N-by-M part of this
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C array contains the input/state matrix B of the transformed
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C system.
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C
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C LDB INTEGER
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C The leading dimension of array B. LDB >= MAX(1,N).
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C
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C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
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C On entry, the leading P-by-N part of this array must
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C contain the original state/output matrix C.
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C On exit, if INFO = 0, the leading P-by-N part of this
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C array contains the state/output matrix C of the
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C transformed system.
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C
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C LDC INTEGER
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C The leading dimension of array C. LDC >= MAX(1,P).
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C
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C NS (output) INTEGER
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C The dimension of the ALPHA-stable subsystem.
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C
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C HSV (output) DOUBLE PRECISION array, dimension (N)
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C If INFO = 0, the leading NS elements of HSV contain the
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C Hankel singular values of the ALPHA-stable part of the
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C original system ordered decreasingly.
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C HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
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C
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C Workspace
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C
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C DWORK DOUBLE PRECISION array, dimension (LDWORK)
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C On exit, if INFO = 0, DWORK(1) returns the optimal value
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C of LDWORK.
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C
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C LDWORK INTEGER
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C The length of the array DWORK.
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C LDWORK >= MAX(1,N*(MAX(N,M,P)+5)+N*(N+1)/2).
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C For optimum performance LDWORK should be larger.
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C
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C Error Indicator
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C
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C INFO INTEGER
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C = 0: successful exit;
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C < 0: if INFO = -i, the i-th argument had an illegal
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C value;
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C = 1: the computation of the ordered real Schur form of A
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C failed;
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C = 2: the separation of the ALPHA-stable/unstable diagonal
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C blocks failed because of very close eigenvalues;
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C = 3: the computed ALPHA-stable part is just stable,
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C having stable eigenvalues very near to the imaginary
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C axis (if DICO = 'C') or to the unit circle
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C (if DICO = 'D');
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C = 4: the computation of Hankel singular values failed.
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C
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C METHOD
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C
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C Let be the following linear system
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C
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C d[x(t)] = Ax(t) + Bu(t)
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C y(t) = Cx(t) (1)
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C
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C where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
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C for a discrete-time system, and let G be the corresponding
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C transfer-function matrix. The following procedure is used to
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C compute the Hankel-norm of the ALPHA-stable projection of G:
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C
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C 1) Decompose additively G as
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C
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C G = G1 + G2
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C
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C such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
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C G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
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C For the computation of the additive decomposition, the
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C algorithm presented in [1] is used.
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C
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C 2) Compute the Hankel-norm of ALPHA-stable projection G1 as the
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C the maximum Hankel singular value of the system (As,Bs,Cs).
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C The computation of the Hankel singular values is performed
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C by using the square-root method of [2].
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C
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C REFERENCES
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C
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C [1] Safonov, M.G., Jonckheere, E.A., Verma, M. and Limebeer, D.J.
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C Synthesis of positive real multivariable feedback systems,
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C Int. J. Control, Vol. 45, pp. 817-842, 1987.
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C
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C [2] Tombs, M.S. and Postlethwaite, I.
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C Truncated balanced realization of stable, non-minimal
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C state-space systems.
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C Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
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C
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C NUMERICAL ASPECTS
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C
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C The implemented method relies on a square-root technique.
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C 3
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C The algorithms require about 17N floating point operations.
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C
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C CONTRIBUTOR
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C
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C C. Oara and A. Varga, German Aerospace Center,
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C DLR Oberpfaffenhofen, July 1998.
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C Based on the RASP routine SHANRM.
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C
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C REVISIONS
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C
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C Nov. 1998, V. Sima, Research Institute for Informatics, Bucharest.
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C Dec. 1998, V. Sima, Katholieke Univ. Leuven, Leuven.
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C Oct. 2001, V. Sima, Research Institute for Informatics, Bucharest.
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C
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C KEYWORDS
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C
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C Additive spectral decomposition, model reduction,
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C multivariable system, state-space model, system norms.
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C
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C ******************************************************************
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C
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C .. Parameters ..
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DOUBLE PRECISION C100, ONE, ZERO
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PARAMETER ( C100 = 100.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
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C .. Scalar Arguments ..
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CHARACTER DICO, EQUIL
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INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NS, P
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DOUBLE PRECISION ALPHA
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C .. Array Arguments ..
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DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
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C .. Local Scalars ..
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LOGICAL DISCR
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INTEGER IERR, KT, KW, KW1, KW2
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DOUBLE PRECISION ALPWRK, MAXRED, WRKOPT
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C .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION AB13AX, DLAMCH
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EXTERNAL AB13AX, DLAMCH, LSAME
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C .. External Subroutines ..
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EXTERNAL TB01ID, TB01KD, XERBLA
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C .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN, SQRT
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C .. Executable Statements ..
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C
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INFO = 0
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DISCR = LSAME( DICO, 'D' )
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C
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C Test the input scalar arguments.
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C
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IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
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INFO = -1
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ELSE IF( .NOT. ( LSAME( EQUIL, 'S' ) .OR.
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$ LSAME( EQUIL, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( M.LT.0 ) THEN
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INFO = -4
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ELSE IF( P.LT.0 ) THEN
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INFO = -5
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ELSE IF( ( DISCR .AND. ( ALPHA.LT.ZERO .OR. ALPHA.GT.ONE ) ) .OR.
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$ ( .NOT.DISCR .AND. ALPHA.GT.ZERO ) ) THEN
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INFO = -6
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -10
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ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
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INFO = -12
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ELSE IF( LDWORK.LT.MAX( 1, N*( MAX( N, M, P ) + 5 ) +
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$ ( N*( N + 1 ) )/2 ) ) THEN
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INFO = -16
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END IF
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C
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IF( INFO.NE.0 ) THEN
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C
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C Error return.
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C
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CALL XERBLA( 'AB13AD', -INFO )
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RETURN
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END IF
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C
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C Quick return if possible.
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C
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IF( MIN( N, M, P ).EQ.0 ) THEN
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NS = 0
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AB13AD = ZERO
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DWORK(1) = ONE
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RETURN
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END IF
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C
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IF( LSAME( EQUIL, 'S' ) ) THEN
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C
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C Scale simultaneously the matrices A, B and C:
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C A <- inv(D)*A*D, B <- inv(D)*B and C <- C*D, where D is a
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C diagonal matrix.
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C Workspace: N.
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C
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MAXRED = C100
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CALL TB01ID( 'All', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
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$ DWORK, INFO )
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END IF
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C
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C Correct the value of ALPHA to ensure stability.
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C
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ALPWRK = ALPHA
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IF( DISCR ) THEN
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IF( ALPHA.EQ.ONE ) ALPWRK = ONE - SQRT( DLAMCH( 'E' ) )
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ELSE
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IF( ALPHA.EQ.ZERO ) ALPWRK = -SQRT( DLAMCH( 'E' ) )
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END IF
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C
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C Allocate working storage.
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C
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KT = 1
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KW1 = N*N + 1
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KW2 = KW1 + N
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KW = KW2 + N
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C
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C Reduce A to a block diagonal real Schur form, with the
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C ALPHA-stable part in the leading diagonal position, using a
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C non-orthogonal similarity transformation A <- inv(T)*A*T and
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C apply the transformation to B and C: B <- inv(T)*B and C <- C*T.
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C
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C Workspace needed: N*(N+2);
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C Additional workspace: need 3*N;
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C prefer larger.
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C
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CALL TB01KD( DICO, 'Stable', 'General', N, M, P, ALPWRK, A, LDA,
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$ B, LDB, C, LDC, NS, DWORK(KT), N, DWORK(KW1),
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$ DWORK(KW2), DWORK(KW), LDWORK-KW+1, IERR )
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IF( IERR.NE.0 ) THEN
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IF( IERR.NE.3 ) THEN
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INFO = 1
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ELSE
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INFO = 2
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END IF
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RETURN
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END IF
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C
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WRKOPT = DWORK(KW) + DBLE( KW-1 )
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C
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IF( NS.EQ.0 ) THEN
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AB13AD = ZERO
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ELSE
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C
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C Workspace: need N*(MAX(N,M,P)+5)+N*(N+1)/2;
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C prefer larger.
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C
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AB13AD = AB13AX( DICO, NS, M, P, A, LDA, B, LDB, C, LDC, HSV,
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$ DWORK, LDWORK, IERR )
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C
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IF( IERR.NE.0 ) THEN
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INFO = IERR + 2
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RETURN
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END IF
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C
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DWORK(1) = MAX( WRKOPT, DWORK(1) )
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END IF
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C
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RETURN
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C *** Last line of AB13AD ***
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END
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