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
C PURPOSE
C
C To compute a reduced order model (Ar,Br,Cr,Dr) for a stable
C original state-space representation (A,B,C,D) by using the optimal
C Hankel-norm approximation method in conjunction with square-root
C balancing. The state dynamics matrix A of the original system is
C an upper quasi-triangular matrix in real Schur canonical form.
C
C ARGUMENTS
C
C Mode Parameters
C
C DICO CHARACTER*1
C Specifies the type of the original system as follows:
C = 'C': continuous-time system;
C = 'D': discrete-time system.
C
C ORDSEL CHARACTER*1
C Specifies the order selection method as follows:
C = 'F': the resulting order NR is fixed;
C = 'A': the resulting order NR is automatically determined
C on basis of the given tolerance TOL1.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the original state-space representation, i.e.
C the 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 NR (input/output) INTEGER
C On entry with ORDSEL = 'F', NR is the desired order of
C the resulting reduced order system. 0 <= NR <= N.
C On exit, if INFO = 0, NR is the order of the resulting
C reduced order model. NR is set as follows:
C if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN),
C where KR is the multiplicity of the Hankel singular value
C HSV(NR+1), NR is the desired order on entry, and NMIN is
C the order of a minimal realization of the given system;
C NMIN is determined as the number of Hankel singular values
C greater than N*EPS*HNORM(A,B,C), where EPS is the machine
C precision (see LAPACK Library Routine DLAMCH) and
C HNORM(A,B,C) is the Hankel norm of the system (computed
C in HSV(1));
C if ORDSEL = 'A', NR is equal to the number of Hankel
C singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)).
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 in a real Schur
C canonical form.
C On exit, if INFO = 0, the leading NR-by-NR part of this
C array contains the state dynamics matrix Ar of the
C reduced order system in a real Schur form.
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 NR-by-M part of this
C array contains the input/state matrix Br of the reduced
C order 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-NR part of this
C array contains the state/output matrix Cr of the reduced
C order system.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C D (input/output) DOUBLE PRECISION array, dimension (LDD,M)
C On entry, the leading P-by-M part of this array must
C contain the original input/output matrix D.
C On exit, if INFO = 0, the leading P-by-M part of this
C array contains the input/output matrix Dr of the reduced
C order system.
C
C LDD INTEGER
C The leading dimension of array D. LDD >= MAX(1,P).
C
C HSV (output) DOUBLE PRECISION array, dimension (N)
C If INFO = 0, it contains the Hankel singular values of
C the original system ordered decreasingly. HSV(1) is the
C Hankel norm of the system.
C
C Tolerances
C
C TOL1 DOUBLE PRECISION
C If ORDSEL = 'A', TOL1 contains the tolerance for
C determining the order of reduced system.
C For model reduction, the recommended value is
C TOL1 = c*HNORM(A,B,C), where c is a constant in the
C interval [0.00001,0.001], and HNORM(A,B,C) is the
C Hankel-norm of the given system (computed in HSV(1)).
C For computing a minimal realization, the recommended
C value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
C machine precision (see LAPACK Library Routine DLAMCH).
C This value is used by default if TOL1 <= 0 on entry.
C If ORDSEL = 'F', the value of TOL1 is ignored.
C
C TOL2 DOUBLE PRECISION
C The tolerance for determining the order of a minimal
C realization of the given system. The recommended value is
C TOL2 = N*EPS*HNORM(A,B,C). This value is used by default
C if TOL2 <= 0 on entry.
C If TOL2 > 0, then TOL2 <= TOL1.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK)
C LIWORK = MAX(1,M), if DICO = 'C';
C LIWORK = MAX(1,N,M), if DICO = 'D'.
C On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
C the computed minimal realization.
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( LDW1,LDW2 ), where
C LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2,
C LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
C MAX( 3*M+1, MIN(N,M)+P ).
C For optimum performance LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 1: with ORDSEL = 'F', the selected order NR is greater
C than the order of a minimal realization of the
C given system. In this case, the resulting NR is set
C automatically to a value corresponding to the order
C of a minimal realization of the system.
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 = 3: the computation of stable projection failed;
C = 4: the order of computed stable projection differs
C from the order of Hankel-norm approximation.
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) + Du(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. The subroutine AB09CX determines for
C the given system (1), the matrices of a reduced order system
C
C d[z(t)] = Ar*z(t) + Br*u(t)
C yr(t) = Cr*z(t) + Dr*u(t) (2)
C
C such that
C
C HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],
C
C where G and Gr are transfer-function matrices of the systems
C (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
C infinity-norm of G.
C
C The optimal Hankel-norm approximation method of [1], based on the
C square-root balancing projection formulas of [2], is employed.
C
C REFERENCES
C
C [1] Glover, K.
C All optimal Hankel norm approximation of linear
C multivariable systems and their L-infinity error bounds.
C Int. J. Control, Vol. 36, pp. 1145-1193, 1984.
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 methods rely on an accuracy enhancing square-root
C technique.
C 3
C The algorithms require less than 30N floating point operations.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center,
C DLR Oberpfaffenhofen, April 1998.
C Based on the RASP routine OHNAP1.
C
C REVISIONS
C
C November 11, 1998, V. Sima, Research Institute for Informatics,
C Bucharest.
C April 24, 2000, A. Varga, DLR Oberpfaffenhofen.
C April 8, 2001, A. Varga, DLR Oberpfaffenhofen.
C March 26, 2005, V. Sima, Research Institute for Informatics.
C
C KEYWORDS
C
C Balancing, Hankel-norm approximation, model reduction,
C multivariable system, state-space model.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
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(*)
C .. Local Scalars
LOGICAL DISCR, FIXORD
INTEGER I, I1, IERR, IRANK, J, KB1, KB2, KC1, KC2T,
$ KHSVP, KHSVP2, KR, KT, KTI, KU, KW, KW1, KW2,
$ LDB1, LDB2, LDC1, LDC2T, NA, NDIM, NKR1, NMINR,
$ NR1, NU, WRKOPT
DOUBLE PRECISION ATOL, RTOL, SKP, SKP2, SRRTOL
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, LSAME
C .. External Subroutines ..
EXTERNAL AB04MD, AB09AX, DAXPY, DCOPY, DGELSY, DGEMM,
$ DLACPY, DSWAP, MA02AD, MB01SD, TB01KD, TB01WD,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
C .. Executable Statements ..
C
INFO = 0
IWARN = 0
DISCR = LSAME( DICO, 'D' )
FIXORD = LSAME( ORDSEL, 'F' )
C
C Check the input scalar arguments.
C
IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
INFO = -1
ELSE IF( .NOT. ( FIXORD .OR. LSAME( ORDSEL, 'A' ) ) ) 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( FIXORD .AND. ( NR.LT.0 .OR. NR.GT.N ) ) 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( LDD.LT.MAX( 1, P ) ) THEN
INFO = -14
ELSE IF( TOL2.GT.ZERO .AND. TOL2.GT.TOL1 ) THEN
INFO = -17
ELSE IF( LDWORK.LT.MAX( N*( 2*N + MAX( N, M, P ) + 5 ) +
$ ( N*( N + 1 ) )/2,
$ N*( M + P + 2 ) + 2*M*P + MIN( N, M ) +
$ MAX ( 3*M + 1, MIN( N, M ) + P ) ) ) THEN
INFO = -20
END IF
C
IF( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'AB09CX', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( MIN( N, M, P ).EQ.0 ) THEN
NR = 0
IWORK(1) = 0
DWORK(1) = ONE
RETURN
END IF
C
RTOL = DBLE( N )*DLAMCH( 'Epsilon' )
SRRTOL = SQRT( RTOL )
C
C Allocate working storage.
C
KT = 1
KTI = KT + N*N
KW = KTI + N*N
C
C Compute a minimal order balanced realization of the given system.
C Workspace: need N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2;
C prefer larger.
C
CALL AB09AX( DICO, 'Balanced', 'Automatic', N, M, P, NMINR, A,
$ LDA, B, LDB, C, LDC, HSV, DWORK(KT), N, DWORK(KTI),
$ N, TOL2, IWORK, DWORK(KW), LDWORK-KW+1, IWARN, INFO )
C
IF( INFO.NE.0 )
$ RETURN
WRKOPT = INT( DWORK(KW) ) + KW - 1
C
C Compute the order of reduced system.
C
ATOL = RTOL*HSV(1)
IF( FIXORD ) THEN
IF( NR.GT.0 ) THEN
IF( NR.GT.NMINR ) THEN
NR = NMINR
IWARN = 1
ENDIF
ENDIF
ELSE
ATOL = MAX( TOL1, ATOL )
NR = 0
DO 10 I = 1, NMINR
IF( HSV(I).LE.ATOL ) GO TO 20
NR = NR + 1
10 CONTINUE
20 CONTINUE
ENDIF
C
IF( NR.EQ.NMINR ) THEN
IWORK(1) = NMINR
DWORK(1) = WRKOPT
KW = N*(N+2)+1
C
C Reduce Ar to a real Schur form.
C
CALL TB01WD( NMINR, M, P, A, LDA, B, LDB, C, LDC,
$ DWORK(2*N+1), N, DWORK, DWORK(N+1), DWORK(KW),
$ LDWORK-KW+1, IERR )
IF( IERR.NE.0 ) THEN
INFO = 3
RETURN
END IF
RETURN
END IF
SKP = HSV(NR+1)
C
C If necessary, reduce the order such that HSV(NR) > HSV(NR+1).
C
30 IF( NR.GT.0 ) THEN
IF( ABS( HSV(NR)-SKP ).LE.SRRTOL*SKP ) THEN
NR = NR - 1
GO TO 30
END IF
END IF
C
C Determine KR, the multiplicity of HSV(NR+1).
C
KR = 1
DO 40 I = NR+2, NMINR
IF( ABS( HSV(I)-SKP ).GT.SRRTOL*SKP ) GO TO 50
KR = KR + 1
40 CONTINUE
50 CONTINUE
C
C For discrete-time case, apply the discrete-to-continuous bilinear
C transformation.
C
IF( DISCR ) THEN
C
C Workspace: need N;
C prefer larger.
C
CALL AB04MD( 'Discrete', NMINR, M, P, ONE, ONE, A, LDA, B, LDB,
$ C, LDC, D, LDD, IWORK, DWORK, LDWORK, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
END IF
C
C Define leading dimensions and offsets for temporary data.
C
NU = NMINR - NR - KR
NA = NR + NU
LDB1 = NA
LDC1 = P
LDB2 = KR
LDC2T = MAX( KR, M )
NR1 = NR + 1
NKR1 = MIN( NMINR, NR1 + KR )
C
KHSVP = 1
KHSVP2 = KHSVP + NA
KU = KHSVP2 + NA
KB1 = KU + P*M
KB2 = KB1 + LDB1*M
KC1 = KB2 + LDB2*M
KC2T = KC1 + LDC1*NA
KW = KC2T + LDC2T*P
C
C Save B2 and C2'.
C
CALL DLACPY( 'Full', KR, M, B(NR1,1), LDB, DWORK(KB2), LDB2 )
CALL MA02AD( 'Full', P, KR, C(1,NR1), LDC, DWORK(KC2T), LDC2T )
IF( NR.GT.0 ) THEN
C
C Permute the elements of HSV and of matrices A, B, C.
C
CALL DCOPY( NR, HSV(1), 1, DWORK(KHSVP), 1 )
CALL DCOPY( NU, HSV(NKR1), 1, DWORK(KHSVP+NR), 1 )
CALL DLACPY( 'Full', NMINR, NU, A(1,NKR1), LDA, A(1,NR1), LDA )
CALL DLACPY( 'Full', NU, NA, A(NKR1,1), LDA, A(NR1,1), LDA )
CALL DLACPY( 'Full', NU, M, B(NKR1,1), LDB, B(NR1,1), LDB )
CALL DLACPY( 'Full', P, NU, C(1,NKR1), LDC, C(1,NR1), LDC )
C
C Save B1 and C1.
C
CALL DLACPY( 'Full', NA, M, B, LDB, DWORK(KB1), LDB1 )
CALL DLACPY( 'Full', P, NA, C, LDC, DWORK(KC1), LDC1 )
END IF
C
C Compute U = C2*pinv(B2').
C Workspace: need N*(M+P+2) + 2*M*P +
C max(min(KR,M)+3*M+1,2*min(KR,M)+P);
C prefer N*(M+P+2) + 2*M*P +
C max(min(KR,M)+2*M+(M+1)*NB,2*min(KR,M)+P*NB),
C where NB is the maximum of the block sizes for
C DGEQP3, DTZRZF, DTZRQF, DORMQR, and DORMRZ.
C
DO 55 J = 1, M
IWORK(J) = 0
55 CONTINUE
CALL DGELSY( KR, M, P, DWORK(KB2), LDB2, DWORK(KC2T), LDC2T,
$ IWORK, RTOL, IRANK, DWORK(KW), LDWORK-KW+1, IERR )
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
CALL MA02AD( 'Full', M, P, DWORK(KC2T), LDC2T, DWORK(KU), P )
C
C Compute D <- D + HSV(NR+1)*U.
C
I = KU
DO 60 J = 1, M
CALL DAXPY( P, SKP, DWORK(I), 1, D(1,J), 1 )
I = I + P
60 CONTINUE
C
IF( NR.GT.0 ) THEN
SKP2 = SKP*SKP
C
C Compute G = inv(S1*S1-skp*skp*I), where S1 is the diagonal
C matrix of relevant singular values (of order NMINR - KR).
C
I1 = KHSVP2
DO 70 I = KHSVP, KHSVP+NA-1
DWORK(I1) = ONE / ( DWORK(I)*DWORK(I) - SKP2 )
I1 = I1 + 1
70 CONTINUE
C
C Compute C <- C1*S1-skp*U*B1'.
C
CALL MB01SD( 'Column', P, NA, C, LDC, DWORK, DWORK(KHSVP) )
CALL DGEMM( 'NoTranspose', 'Transpose', P, NA, M, -SKP,
$ DWORK(KU), P, DWORK(KB1), LDB1, ONE, C, LDC )
C
C Compute B <- G*(S1*B1-skp*C1'*U).
C
CALL MB01SD( 'Row', NA, M, B, LDB, DWORK(KHSVP), DWORK )
CALL DGEMM( 'Transpose', 'NoTranspose', NA, M, P, -SKP,
$ DWORK(KC1), LDC1, DWORK(KU), P, ONE, B, LDB )
CALL MB01SD( 'Row', NA, M, B, LDB, DWORK(KHSVP2), DWORK )
C
C Compute A <- -A1' - B*B1'.
C
DO 80 J = 2, NA
CALL DSWAP( J-1, A(1,J), 1, A(J,1), LDA )
80 CONTINUE
CALL DGEMM( 'NoTranspose', 'Transpose', NA, NA, M, -ONE, B,
$ LDB, DWORK(KB1), LDB1, -ONE, A, LDA )
C
C Extract stable part.
C Workspace: need N*N+5*N;
C prefer larger.
C
KW1 = NA*NA + 1
KW2 = KW1 + NA
KW = KW2 + NA
CALL TB01KD( 'Continuous', 'Stability', 'General', NA, M, P,
$ ZERO, A, LDA, B, LDB, C, LDC, NDIM, DWORK, NA,
$ DWORK(KW1), DWORK(KW2), DWORK(KW), LDWORK-KW+1,
$ IERR )
IF( IERR.NE.0 ) THEN
INFO = 3
RETURN
END IF
C
IF( NDIM.NE.NR ) THEN
INFO = 4
RETURN
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(KW) ) + KW - 1 )
C
C For discrete-time case, apply the continuous-to-discrete
C bilinear transformation.
C
IF( DISCR )
$ CALL AB04MD( 'Continuous', NR, M, P, ONE, ONE, A, LDA, B,
$ LDB, C, LDC, D, LDD, IWORK, DWORK, LDWORK,
$ INFO )
END IF
IWORK(1) = NMINR
DWORK(1) = WRKOPT
C
RETURN
C *** Last line of AB09CX ***
END