SUBROUTINE TB01UD( JOBZ, N, M, P, A, LDA, B, LDB, C, LDC, NCONT,
$ INDCON, NBLK, Z, LDZ, TAU, TOL, IWORK, DWORK,
$ LDWORK, INFO )
C
C PURPOSE
C
C To find a controllable realization for the linear time-invariant
C multi-input system
C
C dX/dt = A * X + B * U,
C Y = C * X,
C
C where A, B, and C are N-by-N, N-by-M, and P-by-N matrices,
C respectively, and A and B are reduced by this routine to
C orthogonal canonical form using (and optionally accumulating)
C orthogonal similarity transformations, which are also applied
C to C. Specifically, the system (A, B, C) is reduced to the
C triplet (Ac, Bc, Cc), where Ac = Z' * A * Z, Bc = Z' * B,
C Cc = C * Z, with
C
C [ Acont * ] [ Bcont ]
C Ac = [ ], Bc = [ ],
C [ 0 Auncont ] [ 0 ]
C
C and
C
C [ A11 A12 . . . A1,p-1 A1p ] [ B1 ]
C [ A21 A22 . . . A2,p-1 A2p ] [ 0 ]
C [ 0 A32 . . . A3,p-1 A3p ] [ 0 ]
C Acont = [ . . . . . . . ], Bc = [ . ],
C [ . . . . . . ] [ . ]
C [ . . . . . ] [ . ]
C [ 0 0 . . . Ap,p-1 App ] [ 0 ]
C
C where the blocks B1, A21, ..., Ap,p-1 have full row ranks and
C p is the controllability index of the pair. The size of the
C block Auncont is equal to the dimension of the uncontrollable
C subspace of the pair (A, B).
C
C ARGUMENTS
C
C Mode Parameters
C
C JOBZ CHARACTER*1
C Indicates whether the user wishes to accumulate in a
C matrix Z the orthogonal similarity transformations for
C reducing the system, as follows:
C = 'N': Do not form Z and do not store the orthogonal
C transformations;
C = 'F': Do not form Z, but store the orthogonal
C transformations in the factored form;
C = 'I': Z is initialized to the unit matrix and the
C orthogonal transformation matrix Z is returned.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the original state-space representation,
C i.e. the order of the matrix A. N >= 0.
C
C M (input) INTEGER
C The number of system inputs, or of columns of B. M >= 0.
C
C P (input) INTEGER
C The number of system outputs, or of rows of C. P >= 0.
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 original state dynamics matrix A.
C On exit, the leading NCONT-by-NCONT part contains the
C upper block Hessenberg state dynamics matrix Acont in Ac,
C given by Z' * A * Z, of a controllable realization for
C the original system. The elements below the first block-
C subdiagonal are set to zero. The leading N-by-N part
C contains the matrix Ac.
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 input matrix B.
C On exit, the leading NCONT-by-M part of this array
C contains the transformed input matrix Bcont in Bc, given
C by Z' * B, with all elements but the first block set to
C zero. The leading N-by-M part contains the matrix Bc.
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 output matrix C.
C On exit, the leading P-by-N part of this array contains
C the transformed output matrix Cc, given by C * Z.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C NCONT (output) INTEGER
C The order of the controllable state-space representation.
C
C INDCON (output) INTEGER
C The controllability index of the controllable part of the
C system representation.
C
C NBLK (output) INTEGER array, dimension (N)
C The leading INDCON elements of this array contain the
C the orders of the diagonal blocks of Acont.
C
C Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
C If JOBZ = 'I', then the leading N-by-N part of this
C array contains the matrix of accumulated orthogonal
C similarity transformations which reduces the given system
C to orthogonal canonical form.
C If JOBZ = 'F', the elements below the diagonal, with the
C array TAU, represent the orthogonal transformation matrix
C as a product of elementary reflectors. The transformation
C matrix can then be obtained by calling the LAPACK Library
C routine DORGQR.
C If JOBZ = 'N', the array Z is not referenced and can be
C supplied as a dummy array (i.e. set parameter LDZ = 1 and
C declare this array to be Z(1,1) in the calling program).
C
C LDZ INTEGER
C The leading dimension of array Z. If JOBZ = 'I' or
C JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
C
C TAU (output) DOUBLE PRECISION array, dimension (N)
C The elements of TAU contain the scalar factors of the
C elementary reflectors used in the reduction of B and A.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used in rank determination when
C transforming (A, B). If the user sets TOL > 0, then
C the given value of TOL is used as a lower bound for the
C reciprocal condition number (see the description of the
C argument RCOND in the SLICOT routine MB03OD); a
C (sub)matrix whose estimated condition number is less than
C 1/TOL is considered to be of full rank. If the user sets
C TOL <= 0, then an implicitly computed, default tolerance,
C defined by TOLDEF = N*N*EPS, is used instead, where EPS
C is the machine precision (see LAPACK Library routine
C DLAMCH).
C
C Workspace
C
C IWORK INTEGER array, dimension (M)
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, 3*M, P).
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
C METHOD
C
C Matrix B is first QR-decomposed and the appropriate orthogonal
C similarity transformation applied to the matrix A. Leaving the
C first rank(B) states unchanged, the remaining lower left block
C of A is then QR-decomposed and the new orthogonal matrix, Q1,
C is also applied to the right of A to complete the similarity
C transformation. By continuing in this manner, a completely
C controllable state-space pair (Acont, Bcont) is found for the
C given (A, B), where Acont is upper block Hessenberg with each
C subdiagonal block of full row rank, and Bcont is zero apart from
C its (independent) first rank(B) rows.
C All orthogonal transformations determined in this process are also
C applied to the matrix C, from the right.
C NOTE that the system controllability indices are easily
C calculated from the dimensions of the blocks of Acont.
C
C REFERENCES
C
C [1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
C Orthogonal Invariants and Canonical Forms for Linear
C Controllable Systems.
C Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
C
C [2] Paige, C.C.
C Properties of numerical algorithms related to computing
C controllablity.
C IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
C
C [3] Petkov, P.Hr., Konstantinov, M.M., Gu, D.W. and
C Postlethwaite, I.
C Optimal Pole Assignment Design of Linear Multi-Input Systems.
C Leicester University, Report 99-11, May 1996.
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(N ) operations and is backward stable.
C
C FURTHER COMMENTS
C
C If the system matrices A and B are badly scaled, it would be
C useful to scale them with SLICOT routine TB01ID, before calling
C the routine.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1998.
C
C REVISIONS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, May 1999, Nov. 2003,
C Mar. 2017.
C A. Varga, DLR Oberpfaffenhofen, March 2002, Nov. 2003.
C
C KEYWORDS
C
C Controllability, minimal realization, orthogonal canonical form,
C orthogonal transformation.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INDCON, INFO, LDA, LDB, LDC, LDWORK, LDZ, M, N,
$ NCONT, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), TAU(*),
$ Z(LDZ,*)
INTEGER IWORK(*), NBLK(*)
C .. Local Scalars ..
LOGICAL LJOBF, LJOBI, LJOBZ
INTEGER IQR, ITAU, J, MCRT, NBL, NCRT, NI, NJ, RANK,
$ WRKOPT
DOUBLE PRECISION ANORM, BNORM, FNRM, TOLDEF
C .. Local Arrays ..
DOUBLE PRECISION SVAL(3)
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE, LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLACPY, DLAPMT, DLASET, DORGQR, DORMQR,
$ MB01PD, MB03OY, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, MIN
C ..
C .. Executable Statements ..
C
INFO = 0
LJOBF = LSAME( JOBZ, 'F' )
LJOBI = LSAME( JOBZ, 'I' )
LJOBZ = LJOBF.OR.LJOBI
C
C Test the input scalar arguments.
C
IF( .NOT.LJOBZ .AND. .NOT.LSAME( JOBZ, 'N' ) ) 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( .NOT.LJOBZ .AND. LDZ.LT.1 .OR.
$ LJOBZ .AND. LDZ.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDWORK.LT.MAX( 1, N, 3*M, P ) ) THEN
INFO = -20
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TB01UD', -INFO )
RETURN
END IF
C
NCONT = 0
INDCON = 0
C
C Calculate the absolute norms of A and B (used for scaling).
C
ANORM = DLANGE( 'M', N, N, A, LDA, DWORK )
BNORM = DLANGE( 'M', N, M, B, LDB, DWORK )
C
C Quick return if possible.
C
IF ( MIN( N, M ).EQ.0 .OR. BNORM.EQ.ZERO ) THEN
IF( N.GT.0 ) THEN
IF ( LJOBI ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
ELSE IF ( LJOBF ) THEN
CALL DLASET( 'Full', N, N, ZERO, ZERO, Z, LDZ )
CALL DLASET( 'Full', N, 1, ZERO, ZERO, TAU, N )
END IF
END IF
DWORK(1) = ONE
RETURN
END IF
C
C Scale (if needed) the matrices A and B.
C
CALL MB01PD( 'S', 'G', N, N, 0, 0, ANORM, 0, NBLK, A, LDA, INFO )
CALL MB01PD( 'S', 'G', N, M, 0, 0, BNORM, 0, NBLK, B, LDB, INFO )
C
C Compute the Frobenius norm of [ B A ] (used for rank estimation).
C
FNRM = DLANGE( 'F', N, M, B, LDB, DWORK )
C
TOLDEF = TOL
IF ( TOLDEF.LE.ZERO ) THEN
C
C Use the default tolerance in controllability determination.
C
TOLDEF = DBLE( N*N )*DLAMCH( 'Precision' )
END IF
C
IF ( FNRM.LT.TOLDEF )
$ FNRM = ONE
C
WRKOPT = 1
NI = 0
ITAU = 1
NCRT = N
MCRT = M
IQR = 1
C
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of real workspace needed at that point in the
C code, as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately
C following subroutine, as returned by ILAENV.)
C
10 CONTINUE
C
C Rank-revealing QR decomposition with column pivoting.
C The calculation is performed in NCRT rows of B starting from
C the row IQR (initialized to 1 and then set to rank(B)+1).
C Workspace: 3*MCRT.
C
CALL MB03OY( NCRT, MCRT, B(IQR,1), LDB, TOLDEF, FNRM, RANK,
$ SVAL, IWORK, TAU(ITAU), DWORK, INFO )
C
IF ( RANK.NE.0 ) THEN
NJ = NI
NI = NCONT
NCONT = NCONT + RANK
INDCON = INDCON + 1
NBLK(INDCON) = RANK
C
C Premultiply and postmultiply the appropriate block row
C and block column of A by Q' and Q, respectively.
C Workspace: need NCRT;
C prefer NCRT*NB.
C
CALL DORMQR( 'Left', 'Transpose', NCRT, NCRT, RANK,
$ B(IQR,1), LDB, TAU(ITAU), A(NI+1,NI+1), LDA,
$ DWORK, LDWORK, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
C
C Workspace: need N;
C prefer N*NB.
C
CALL DORMQR( 'Right', 'No transpose', N, NCRT, RANK,
$ B(IQR,1), LDB, TAU(ITAU), A(1,NI+1), LDA,
$ DWORK, LDWORK, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
C
C Postmultiply the appropriate block column of C by Q.
C Workspace: need P;
C prefer P*NB.
C
CALL DORMQR( 'Right', 'No transpose', P, NCRT, RANK,
$ B(IQR,1), LDB, TAU(ITAU), C(1,NI+1), LDC,
$ DWORK, LDWORK, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
C
C If required, save transformations.
C
IF ( LJOBZ.AND.NCRT.GT.1 ) THEN
CALL DLACPY( 'L', NCRT-1, MIN( RANK, NCRT-1 ),
$ B(IQR+1,1), LDB, Z(NI+2,ITAU), LDZ )
END IF
C
C Zero the subdiagonal elements of the current matrix.
C
IF ( RANK.GT.1 )
$ CALL DLASET( 'L', RANK-1, RANK-1, ZERO, ZERO, B(IQR+1,1),
$ LDB )
C
C Backward permutation of the columns of B or A.
C
IF ( INDCON.EQ.1 ) THEN
CALL DLAPMT( .FALSE., RANK, M, B(IQR,1), LDB, IWORK )
IQR = RANK + 1
FNRM = DLANGE( 'F', N, N, A, LDA, DWORK )
ELSE
DO 20 J = 1, MCRT
CALL DCOPY( RANK, B(IQR,J), 1, A(NI+1,NJ+IWORK(J)),
$ 1 )
20 CONTINUE
END IF
C
ITAU = ITAU + RANK
IF ( RANK.NE.NCRT ) THEN
MCRT = RANK
NCRT = NCRT - RANK
CALL DLACPY( 'G', NCRT, MCRT, A(NCONT+1,NI+1), LDA,
$ B(IQR,1), LDB )
CALL DLASET( 'G', NCRT, MCRT, ZERO, ZERO,
$ A(NCONT+1,NI+1), LDA )
GO TO 10
END IF
END IF
C
C If required, accumulate transformations.
C Workspace: need N; prefer N*NB.
C
IF ( LJOBI ) THEN
CALL DORGQR( N, N, ITAU-1, Z, LDZ, TAU, DWORK,
$ LDWORK, INFO )
WRKOPT = MAX( WRKOPT, INT( DWORK(1) ) )
END IF
C
C Annihilate the trailing blocks of B.
C
IF( IQR.LE.N )
$ CALL DLASET( 'G', N-IQR+1, M, ZERO, ZERO, B(IQR,1), LDB )
C
C Annihilate the trailing elements of TAU, if JOBZ = 'F'.
C
IF ( LJOBF ) THEN
DO 30 J = ITAU, N
TAU(J) = ZERO
30 CONTINUE
END IF
C
C Undo scaling of A and B.
C
IF ( INDCON.LT.N ) THEN
NBL = INDCON + 1
NBLK(NBL) = N - NCONT
ELSE
NBL = 0
END IF
CALL MB01PD( 'U', 'H', N, N, 0, 0, ANORM, NBL, NBLK, A, LDA,
$ INFO )
CALL MB01PD( 'U', 'G', NBLK(1), M, 0, 0, BNORM, 0, NBLK, B, LDB,
$ INFO )
C
C Set optimal workspace dimension.
C
DWORK(1) = WRKOPT
RETURN
C *** Last line of TB01UD ***
END