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DOUBLE PRECISION FUNCTION AB13AX( DICO, N, M, P, A, LDA, B, LDB,
$ C, LDC, HSV, DWORK, LDWORK,
$ INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To compute the Hankel-norm of the transfer-function matrix G of
C a stable state-space system (A,B,C). The state dynamics matrix A
C of the given system is an upper quasi-triangular matrix in
C real Schur form.
C
C FUNCTION VALUE
C
C AB13AX DOUBLE PRECISION
C The Hankel-norm of G (if INFO = 0).
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of the system as follows:
C = 'C': continuous-time system;
C = 'D': discrete-time system.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the state-space representation, i.e. the
C order of the matrix A. N >= 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C P (input) INTEGER
C The number of system outputs. P >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C state dynamics matrix A in a real Schur canonical form.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C input/state matrix B.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading P-by-N part of this array must contain the
C state/output matrix C.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C HSV (output) DOUBLE PRECISION array, dimension (N)
C If INFO = 0, this array contains the Hankel singular
C values of the given system ordered decreasingly.
C HSV(1) is the Hankel norm of the given system.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX(1,N*(MAX(N,M,P)+5)+N*(N+1)/2).
C For optimum performance LDWORK should be larger.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 1: the state matrix A is not stable (if DICO = 'C')
C or not convergent (if DICO = 'D');
C = 2: the computation of Hankel singular values failed.
C
C METHOD
C
C Let be the stable linear system
C
C d[x(t)] = Ax(t) + Bu(t)
C y(t) = Cx(t) (1)
C
C where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
C for a discrete-time system, and let G be the corresponding
C transfer-function matrix. The Hankel-norm of G is computed as the
C the maximum Hankel singular value of the system (A,B,C).
C The computation of the Hankel singular values is performed
C by using the square-root method of [1].
C
C REFERENCES
C
C [1] Tombs M.S. and Postlethwaite I.
C Truncated balanced realization of stable, non-minimal
C state-space systems.
C Int. J. Control, Vol. 46, pp. 1319-1330, 1987.
C
C NUMERICAL ASPECTS
C
C The implemented method relies on a square-root technique.
C 3
C The algorithms require about 17N floating point operations.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center,
C DLR Oberpfaffenhofen, July 1998.
C Based on the RASP routine SHANRM.
C
C REVISIONS
C
C Nov. 1998, V. Sima, Research Institute for Informatics, Bucharest.
C Feb. 2000, V. Sima, Research Institute for Informatics, Bucharest.
C Oct. 2001, V. Sima, Research Institute for Informatics, Bucharest.
C
C KEYWORDS
C
C Multivariable system, state-space model, system norms.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
C .. Local Scalars ..
LOGICAL DISCR
INTEGER I, IERR, J, KR, KS, KTAU, KU, KW, MNMP
DOUBLE PRECISION SCALEC, SCALEO, WRKOPT
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DLACPY, DSCAL, DTPMV, MA02DD, MB03UD, SB03OU,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
DISCR = LSAME( DICO, 'D' )
C
C Test the input scalar arguments.
C
IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( P.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -10
ELSE IF( LDWORK.LT.MAX( 1, N*( MAX( N, M, P ) + 5 ) +
$ ( N*( N + 1 ) )/2 ) ) THEN
INFO = -13
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB13AX', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 ) THEN
AB13AX = ZERO
DWORK(1) = ONE
RETURN
END IF
C
C Allocate N*MAX(N,M,P), N, and N*(N+1)/2 working storage for the
C matrices S, TAU, and R, respectively. S shares the storage with U.
C
KU = 1
KS = 1
MNMP = MAX( N, M, P )
KTAU = KS + N*MNMP
KR = KTAU + N
KW = KR
C
C Copy C in U.
C
CALL DLACPY( 'Full', P, N, C, LDC, DWORK(KU), MNMP )
C
C If DISCR = .FALSE., solve for R the Lyapunov equation
C 2
C A'*(R'*R) + (R'*R)*A + scaleo * C'*C = 0 .
C
C If DISCR = .TRUE., solve for R the Lyapunov equation
C 2
C A'*(R'*R)*A + scaleo * C'*C = R'*R .
C
C Workspace needed: N*(MAX(N,M,P)+1);
C Additional workspace: need 4*N;
C prefer larger.
C
CALL SB03OU( DISCR, .FALSE., N, P, A, LDA, DWORK(KU), MNMP,
$ DWORK(KTAU), DWORK(KU), N, SCALEO, DWORK(KW),
$ LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
INFO = 1
RETURN
ENDIF
C
WRKOPT = DWORK(KW) + DBLE( KW-1 )
C
C Pack the upper triangle of R in DWORK(KR).
C Workspace needed: N*(MAX(N,M,P) + 1) + N*(N+1)/2.
C
CALL MA02DD( 'Pack', 'Upper', N, DWORK(KU), N, DWORK(KR) )
C
KW = KR + ( N*( N + 1 ) )/2
C
C Copy B in S (over U).
C
CALL DLACPY( 'Full', N, M, B, LDB, DWORK(KS), N )
C
C If DISCR = .FALSE., solve for S the Lyapunov equation
C 2
C A*(S*S') + (S*S')*A' + scalec *B*B' = 0 .
C
C If DISCR = .TRUE., solve for S the Lyapunov equation
C 2
C A*(S*S')*A' + scalec *B*B' = S*S' .
C
C Workspace needed: N*(MAX(N,M,P) + 1) + N*(N+1)/2;
C Additional workspace: need 4*N;
C prefer larger.
C
CALL SB03OU( DISCR, .TRUE., N, M, A, LDA, DWORK(KS), N,
$ DWORK(KTAU), DWORK(KS), N, SCALEC, DWORK(KW),
$ LDWORK-KW+1, IERR )
C
WRKOPT = MAX( WRKOPT, DWORK(KW) + DBLE( KW-1 ) )
C
C | x x |
C Compute R*S in the form | 0 x | in S. Note that R is packed.
C
J = KS
DO 10 I = 1, N
CALL DTPMV( 'Upper', 'NoTranspose', 'NonUnit', I, DWORK(KR),
$ DWORK(J), 1 )
J = J + N
10 CONTINUE
C
C Compute the singular values of the upper triangular matrix R*S.
C
C Workspace needed: N*MAX(N,M,P);
C Additional workspace: need MAX(1,5*N);
C prefer larger.
C
KW = KTAU
CALL MB03UD( 'NoVectors', 'NoVectors', N, DWORK(KS), N, DWORK, 1,
$ HSV, DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
INFO = 2
RETURN
ENDIF
C
C Scale singular values.
C
CALL DSCAL( N, ONE / SCALEC / SCALEO, HSV, 1 )
AB13AX = HSV(1)
C
DWORK(1) = MAX( WRKOPT, DWORK(KW) + DBLE( KW-1 ) )
C
RETURN
C *** Last line of AB13AX ***
END

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