SUBROUTINE TB01PD( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC,
$ NR, TOL, IWORK, DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To find a reduced (controllable, observable, or minimal) state-
C space representation (Ar,Br,Cr) for any original state-space
C representation (A,B,C). The matrix Ar is in upper block
C Hessenberg form.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOB CHARACTER*1
C Indicates whether the user wishes to remove the
C uncontrollable and/or unobservable parts as follows:
C = 'M': Remove both the uncontrollable and unobservable
C parts to get a minimal state-space representation;
C = 'C': Remove the uncontrollable part only to get a
C controllable state-space representation;
C = 'O': Remove the unobservable part only to get an
C observable state-space representation.
C
C EQUIL CHARACTER*1
C Specifies whether the user wishes to preliminarily balance
C the triplet (A,B,C) as follows:
C = 'S': Perform balancing (scaling);
C = 'N': Do not perform balancing.
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 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 NR-by-NR part of this array contains
C the upper block Hessenberg state dynamics matrix Ar of a
C minimal, controllable, or observable realization for the
C original system, depending on the value of JOB, JOB = 'M',
C JOB = 'C', or JOB = 'O', respectively.
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 if JOB = 'C', or (LDB,MAX(M,P)), otherwise.
C On entry, the leading N-by-M part of this array must
C contain the original input/state matrix B; if JOB = 'M',
C or JOB = 'O', the remainder of the leading N-by-MAX(M,P)
C part is used as internal workspace.
C On exit, the leading NR-by-M part of this array contains
C the transformed input/state matrix Br of a minimal,
C controllable, or observable realization for the original
C system, depending on the value of JOB, JOB = 'M',
C JOB = 'C', or JOB = 'O', respectively.
C If JOB = 'C', only the first IWORK(1) rows of B are
C nonzero.
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; if JOB = 'M',
C or JOB = 'O', the remainder of the leading MAX(M,P)-by-N
C part is used as internal workspace.
C On exit, the leading P-by-NR part of this array contains
C the transformed state/output matrix Cr of a minimal,
C controllable, or observable realization for the original
C system, depending on the value of JOB, JOB = 'M',
C JOB = 'C', or JOB = 'O', respectively.
C If JOB = 'M', or JOB = 'O', only the last IWORK(1) columns
C (in the first NR columns) of C are nonzero.
C
C LDC INTEGER
C The leading dimension of array C.
C LDC >= MAX(1,M,P) if N > 0.
C LDC >= 1 if N = 0.
C
C NR (output) INTEGER
C The order of the reduced state-space representation
C (Ar,Br,Cr) of a minimal, controllable, or observable
C realization for the original system, depending on
C JOB = 'M', JOB = 'C', or JOB = 'O'.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used in rank determination when
C transforming (A, B, C). 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 (determined by the SLICOT routine TB01UD) is used instead.
C
C Workspace
C
C IWORK INTEGER array, dimension (N+MAX(M,P))
C On exit, if INFO = 0, the first nonzero elements of
C IWORK(1:N) return the orders of the diagonal blocks of A.
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, 3*M, 3*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 If JOB = 'M', the matrices A and B are operated on by orthogonal
C similarity transformations (made up of products of Householder
C transformations) so as to produce an upper block Hessenberg matrix
C A1 and a matrix B1 with all but its first rank(B) rows zero; this
C separates out the controllable part of the original system.
C Applying the same algorithm to the dual of this subsystem,
C therefore separates out the controllable and observable (i.e.
C minimal) part of the original system representation, with the
C final Ar upper block Hessenberg (after using pertransposition).
C If JOB = 'C', or JOB = 'O', only the corresponding part of the
C above procedure is applied.
C
C REFERENCES
C
C [1] Van Dooren, P.
C The Generalized Eigenstructure Problem in Linear System
C Theory. (Algorithm 1)
C IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(N ) operations and is backward stable.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1998.
C
C REVISIONS
C
C A. Varga, DLR Oberpfaffenhofen, July 1998.
C A. Varga, DLR Oberpfaffenhofen, April 28, 1999.
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2004.
C
C KEYWORDS
C
C Hessenberg form, minimal realization, multivariable system,
C orthogonal transformation, state-space model, state-space
C representation.
C
C ******************************************************************
C
C .. Parameters ..
INTEGER LDIZ
PARAMETER ( LDIZ = 1 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER EQUIL, JOB
INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)
C .. Local Scalars ..
LOGICAL LEQUIL, LNJOBC, LNJOBO
INTEGER I, INDCON, ITAU, IZ, JWORK, KL, MAXMP, NCONT,
$ WRKOPT
DOUBLE PRECISION MAXRED
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL AB07MD, TB01ID, TB01UD, TB01XD, XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
MAXMP = MAX( M, P )
LNJOBC = .NOT.LSAME( JOB, 'C' )
LNJOBO = .NOT.LSAME( JOB, 'O' )
LEQUIL = LSAME( EQUIL, 'S' )
C
C Test the input scalar arguments.
C
IF( LNJOBC .AND. LNJOBO .AND. .NOT.LSAME( JOB, 'M' ) ) THEN
INFO = -1
ELSE IF( .NOT.LEQUIL .AND. .NOT.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( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.1 .OR. ( N.GT.0 .AND. LDC.LT.MAXMP ) ) THEN
INFO = -11
ELSE IF( LDWORK.LT.MAX( 1, N + MAX( N, 3*MAXMP ) ) ) THEN
INFO = -16
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TB01PD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( N.EQ.0 .OR. ( LNJOBC .AND. MIN( N, P ).EQ.0 ) .OR.
$ ( LNJOBO .AND. MIN( N, M ).EQ.0 ) ) THEN
NR = 0
C
DO 5 I = 1, N
IWORK(I) = 0
5 CONTINUE
C
DWORK(1) = ONE
RETURN
END IF
C
C If required, balance the triplet (A,B,C) (default MAXRED).
C Workspace: need N.
C
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of real workspace needed at that point in the code,
C as well as the preferred amount for good performance.)
C
IF ( LEQUIL ) THEN
MAXRED = ZERO
CALL TB01ID( 'A', N, M, P, MAXRED, A, LDA, B, LDB, C, LDC,
$ DWORK, INFO )
WRKOPT = N
ELSE
WRKOPT = 1
END IF
C
IZ = 1
ITAU = 1
JWORK = ITAU + N
IF ( LNJOBO ) THEN
C
C Separate out controllable subsystem (of order NCONT):
C A <-- Z'*A*Z, B <-- Z'*B, C <-- C*Z.
C
C Workspace: need N + MAX(N, 3*M, P).
C prefer larger.
C
CALL TB01UD( 'No Z', N, M, P, A, LDA, B, LDB, C, LDC, NCONT,
$ INDCON, IWORK, DWORK(IZ), LDIZ, DWORK(ITAU), TOL,
$ IWORK(N+1), DWORK(JWORK), LDWORK-JWORK+1, INFO )
C
WRKOPT = INT( DWORK(JWORK) ) + JWORK - 1
ELSE
NCONT = N
END IF
C
IF ( LNJOBC ) THEN
C
C Separate out the observable subsystem (of order NR):
C Form the dual of the subsystem of order NCONT (which is
C controllable, if JOB = 'M'), leaving rest as it is.
C
CALL AB07MD( 'Z', NCONT, M, P, A, LDA, B, LDB, C, LDC, DWORK,
$ 1, INFO )
C
C And separate out the controllable part of this dual subsystem.
C
C Workspace: need NCONT + MAX(NCONT, 3*P, M).
C prefer larger.
C
CALL TB01UD( 'No Z', NCONT, P, M, A, LDA, B, LDB, C, LDC, NR,
$ INDCON, IWORK, DWORK(IZ), LDIZ, DWORK(ITAU), TOL,
$ IWORK(N+1), DWORK(JWORK), LDWORK-JWORK+1, INFO )
C
WRKOPT = MAX( WRKOPT, INT( DWORK(JWORK) )+JWORK-1 )
C
C Transpose and reorder (to get a block upper Hessenberg
C matrix A), giving, for JOB = 'M', the controllable and
C observable (i.e., minimal) part of original system.
C
IF( INDCON.GT.0 ) THEN
KL = IWORK(1) - 1
IF ( INDCON.GE.2 )
$ KL = KL + IWORK(2)
ELSE
KL = 0
END IF
CALL TB01XD( 'Zero D', NR, P, M, KL, MAX( 0, NR-1 ), A, LDA,
$ B, LDB, C, LDC, DWORK, 1, INFO )
ELSE
NR = NCONT
END IF
C
C Annihilate the trailing components of IWORK(1:N).
C
DO 10 I = INDCON + 1, N
IWORK(I) = 0
10 CONTINUE
C
C Set optimal workspace dimension.
C
DWORK(1) = WRKOPT
RETURN
C *** Last line of TB01PD ***
END