DOUBLE PRECISION FUNCTION AB13AD( DICO, EQUIL, N, M, P, ALPHA, A,
$ LDA, B, LDB, C, LDC, NS, HSV,
$ DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To compute the Hankel-norm of the ALPHA-stable projection of the
C transfer-function matrix G of the state-space system (A,B,C).
C
C FUNCTION VALUE
C
C AB13AD DOUBLE PRECISION
C The Hankel-norm of the ALPHA-stable projection of G
C (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 EQUIL CHARACTER*1
C Specifies whether the user wishes to preliminarily
C equilibrate the triplet (A,B,C) as follows:
C = 'S': perform equilibration (scaling);
C = 'N': do not perform equilibration.
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 ALPHA (input) DOUBLE PRECISION
C Specifies the ALPHA-stability boundary for the eigenvalues
C of the state dynamics matrix A. For a continuous-time
C system (DICO = 'C'), ALPHA <= 0 is the boundary value for
C the real parts of eigenvalues, while for a discrete-time
C system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
C boundary value for the moduli of eigenvalues.
C The ALPHA-stability domain does not include the boundary
C (see the Note below).
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the state dynamics matrix A.
C On exit, if INFO = 0, the leading N-by-N part of this
C array contains the state dynamics matrix A in a block
C diagonal real Schur form with its eigenvalues reordered
C and separated. The resulting A has two diagonal blocks.
C The leading NS-by-NS part of A has eigenvalues in the
C ALPHA-stability domain and the trailing (N-NS) x (N-NS)
C part has eigenvalues outside the ALPHA-stability domain.
C Note: The ALPHA-stability domain is defined either
C as the open half complex plane left to ALPHA,
C for a continous-time system (DICO = 'C'), or the
C interior of the ALPHA-radius circle centered in the
C origin, for a discrete-time system (DICO = 'D').
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
C On entry, the leading N-by-M part of this array must
C contain the original input/state matrix B.
C On exit, if INFO = 0, the leading N-by-M part of this
C array contains the input/state matrix B of the transformed
C system.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the original state/output matrix C.
C On exit, if INFO = 0, the leading P-by-N part of this
C array contains the state/output matrix C of the
C transformed system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C NS (output) INTEGER
C The dimension of the ALPHA-stable subsystem.
C
C HSV (output) DOUBLE PRECISION array, dimension (N)
C If INFO = 0, the leading NS elements of HSV contain the
C Hankel singular values of the ALPHA-stable part of the
C original system ordered decreasingly.
C HSV(1) is the Hankel norm of the ALPHA-stable subsystem.
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 computation of the ordered real Schur form of A
C failed;
C = 2: the separation of the ALPHA-stable/unstable diagonal
C blocks failed because of very close eigenvalues;
C = 3: the computed ALPHA-stable part is just stable,
C having stable eigenvalues very near to the imaginary
C axis (if DICO = 'C') or to the unit circle
C (if DICO = 'D');
C = 4: the computation of Hankel singular values failed.
C
C METHOD
C
C Let be the following 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 following procedure is used to
C compute the Hankel-norm of the ALPHA-stable projection of G:
C
C 1) Decompose additively G as
C
C G = G1 + G2
C
C such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
C G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
C For the computation of the additive decomposition, the
C algorithm presented in [1] is used.
C
C 2) Compute the Hankel-norm of ALPHA-stable projection G1 as the
C the maximum Hankel singular value of the system (As,Bs,Cs).
C The computation of the Hankel singular values is performed
C by using the square-root method of [2].
C
C REFERENCES
C
C [1] Safonov, M.G., Jonckheere, E.A., Verma, M. and Limebeer, D.J.
C Synthesis of positive real multivariable feedback systems,
C Int. J. Control, Vol. 45, pp. 817-842, 1987.
C
C [2] 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, DLR Oberpfaffenhofen,
C July 1998.
C Based on the RASP routine SHANRM.
C
C REVISIONS
C
C Nov. 1998, V. Sima, Research Institute for Informatics, Bucharest.
C Dec. 1998, V. Sima, Katholieke Univ. Leuven, Leuven.
C Oct. 2001, V. Sima, Research Institute for Informatics, Bucharest.
C
C KEYWORDS
C
C Additive spectral decomposition, model reduction,
C multivariable system, state-space model, system norms.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION C100, ONE, ZERO
PARAMETER ( C100 = 100.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NS, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)
C .. Local Scalars ..
LOGICAL DISCR
INTEGER IERR, KT, KW, KW1, KW2
DOUBLE PRECISION ALPWRK, MAXRED, WRKOPT
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION AB13AX, DLAMCH
EXTERNAL AB13AX, DLAMCH, LSAME
C .. External Subroutines ..
EXTERNAL TB01ID, TB01KD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN, SQRT
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( .NOT. ( LSAME( EQUIL, 'S' ) .OR.
$ LSAME( EQUIL, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( ( DISCR .AND. ( ALPHA.LT.ZERO .OR. ALPHA.GT.ONE ) ) .OR.
$ ( .NOT.DISCR .AND. ALPHA.GT.ZERO ) ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDWORK.LT.MAX( 1, N*( MAX( N, M, P ) + 5 ) +
$ ( N*( N + 1 ) )/2 ) ) THEN
INFO = -16
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB13AD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 ) THEN
NS = 0
AB13AD = ZERO
DWORK(1) = ONE
RETURN
END IF
C
IF( LSAME( EQUIL, 'S' ) ) THEN
C
C Scale simultaneously the matrices A, B and C:
C A <- inv(D)*A*D, B <- inv(D)*B and C <- C*D, where D is a
C diagonal matrix.
C Workspace: N.
C
MAXRED = C100
CALL TB01ID( 'All', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
$ DWORK, INFO )
END IF
C
C Correct the value of ALPHA to ensure stability.
C
ALPWRK = ALPHA
IF( DISCR ) THEN
IF( ALPHA.EQ.ONE ) ALPWRK = ONE - SQRT( DLAMCH( 'E' ) )
ELSE
IF( ALPHA.EQ.ZERO ) ALPWRK = -SQRT( DLAMCH( 'E' ) )
END IF
C
C Allocate working storage.
C
KT = 1
KW1 = N*N + 1
KW2 = KW1 + N
KW = KW2 + N
C
C Reduce A to a block diagonal real Schur form, with the
C ALPHA-stable part in the leading diagonal position, using a
C non-orthogonal similarity transformation A <- inv(T)*A*T and
C apply the transformation to B and C: B <- inv(T)*B and C <- C*T.
C
C Workspace needed: N*(N+2);
C Additional workspace: need 3*N;
C prefer larger.
C
CALL TB01KD( DICO, 'Stable', 'General', N, M, P, ALPWRK, A, LDA,
$ B, LDB, C, LDC, NS, DWORK(KT), N, DWORK(KW1),
$ DWORK(KW2), DWORK(KW), LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
IF( IERR.NE.3 ) THEN
INFO = 1
ELSE
INFO = 2
END IF
RETURN
END IF
C
WRKOPT = DWORK(KW) + DBLE( KW-1 )
C
IF( NS.EQ.0 ) THEN
AB13AD = ZERO
ELSE
C
C Workspace: need N*(MAX(N,M,P)+5)+N*(N+1)/2;
C prefer larger.
C
AB13AD = AB13AX( DICO, NS, M, P, A, LDA, B, LDB, C, LDC, HSV,
$ DWORK, LDWORK, IERR )
C
IF( IERR.NE.0 ) THEN
INFO = IERR + 2
RETURN
END IF
C
DWORK(1) = MAX( WRKOPT, DWORK(1) )
END IF
C
RETURN
C *** Last line of AB13AD ***
END