SUBROUTINE TD04AD( ROWCOL, M, P, INDEX, DCOEFF, LDDCOE, UCOEFF,
$ LDUCO1, LDUCO2, NR, A, LDA, B, LDB, C, LDC, D,
$ LDD, TOL, IWORK, DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To find a minimal state-space representation (A,B,C,D) for a
C proper transfer matrix T(s) given as either row or column
C polynomial vectors over denominator polynomials, possibly with
C uncancelled common terms.
C
C ARGUMENTS
C
C Mode Parameters
C
C ROWCOL CHARACTER*1
C Indicates whether the transfer matrix T(s) is given as
C rows or columns over common denominators as follows:
C = 'R': T(s) is given as rows over common denominators;
C = 'C': T(s) is given as columns over common denominators.
C
C Input/Output Parameters
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 INDEX (input) INTEGER array, dimension (porm), where porm = P,
C if ROWCOL = 'R', and porm = M, if ROWCOL = 'C'.
C This array must contain the degrees of the denominator
C polynomials in D(s).
C
C DCOEFF (input) DOUBLE PRECISION array, dimension (LDDCOE,kdcoef),
C where kdcoef = MAX(INDEX(I)) + 1.
C The leading porm-by-kdcoef part of this array must contain
C the coefficients of each denominator polynomial.
C DCOEFF(I,K) is the coefficient in s**(INDEX(I)-K+1) of the
C I-th denominator polynomial in D(s), where
C K = 1,2,...,kdcoef.
C
C LDDCOE INTEGER
C The leading dimension of array DCOEFF.
C LDDCOE >= MAX(1,P) if ROWCOL = 'R';
C LDDCOE >= MAX(1,M) if ROWCOL = 'C'.
C
C UCOEFF (input) DOUBLE PRECISION array, dimension
C (LDUCO1,LDUCO2,kdcoef)
C The leading P-by-M-by-kdcoef part of this array must
C contain the numerator matrix U(s); if ROWCOL = 'C', this
C array is modified internally but restored on exit, and the
C remainder of the leading MAX(M,P)-by-MAX(M,P)-by-kdcoef
C part is used as internal workspace.
C UCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
C of polynomial (I,J) of U(s), where K = 1,2,...,kdcoef;
C if ROWCOL = 'R' then iorj = I, otherwise iorj = J.
C Thus for ROWCOL = 'R', U(s) =
C diag(s**INDEX(I))*(UCOEFF(.,.,1)+UCOEFF(.,.,2)/s+...).
C
C LDUCO1 INTEGER
C The leading dimension of array UCOEFF.
C LDUCO1 >= MAX(1,P) if ROWCOL = 'R';
C LDUCO1 >= MAX(1,M,P) if ROWCOL = 'C'.
C
C LDUCO2 INTEGER
C The second dimension of array UCOEFF.
C LDUCO2 >= MAX(1,M) if ROWCOL = 'R';
C LDUCO2 >= MAX(1,M,P) if ROWCOL = 'C'.
C
C NR (output) INTEGER
C The order of the resulting minimal realization, i.e. the
C order of the state dynamics matrix A.
C
C A (output) DOUBLE PRECISION array, dimension (LDA,N),
C porm
C where N = SUM INDEX(I).
C I=1
C The leading NR-by-NR part of this array contains the upper
C block Hessenberg state dynamics matrix A of a minimal
C realization.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C B (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
C The leading NR-by-M part of this array contains the
C input/state matrix B of a minimal realization; the
C remainder of the leading N-by-MAX(M,P) part is used as
C internal workspace.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (output) DOUBLE PRECISION array, dimension (LDC,N)
C The leading P-by-NR part of this array contains the
C state/output matrix C of a minimal realization; the
C remainder of the leading MAX(M,P)-by-N part is used as
C internal workspace.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,M,P).
C
C D (output) DOUBLE PRECISION array, dimension (LDD,M),
C if ROWCOL = 'R', and (LDD,MAX(M,P)) if ROWCOL = 'C'.
C The leading P-by-M part of this array contains the direct
C transmission matrix D; if ROWCOL = 'C', the remainder of
C the leading MAX(M,P)-by-MAX(M,P) part is used as internal
C workspace.
C
C LDD INTEGER
C The leading dimension of array D.
C LDD >= MAX(1,P) if ROWCOL = 'R';
C LDD >= MAX(1,M,P) if ROWCOL = 'C'.
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 > 0: if INFO = i, then i is the first integer for which
C ABS( DCOEFF(I,1) ) is so small that the calculations
C would overflow (see SLICOT Library routine TD03AY);
C that is, the leading coefficient of a polynomial is
C nearly zero; no state-space representation is
C calculated.
C
C METHOD
C
C The method for transfer matrices factorized by rows will be
C described here: T(s) factorized by columns is dealt with by
C operating on the dual T'(s). This description for T(s) is
C actually the left polynomial matrix representation
C
C T(s) = inv(D(s))*U(s),
C
C where D(s) is diagonal with its (I,I)-th polynomial element of
C degree INDEX(I). The first step is to check whether the leading
C coefficient of any polynomial element of D(s) is approximately
C zero; if so the routine returns with INFO > 0. Otherwise,
C Wolovich's Observable Structure Theorem is used to construct a
C state-space representation in observable companion form which
C is equivalent to the above polynomial matrix representation.
C The method is particularly easy here due to the diagonal form
C of D(s). This state-space representation is not necessarily
C controllable (as D(s) and U(s) are not necessarily relatively
C left prime), but it is in theory completely observable; however,
C its observability matrix may be poorly conditioned, so it is
C treated as a general state-space representation and SLICOT
C Library routine TB01PD is then called to separate out a minimal
C realization from this general state-space representation by means
C of orthogonal similarity transformations.
C
C REFERENCES
C
C [1] Patel, R.V.
C Computation of Minimal-Order State-Space Realizations and
C Observability Indices using Orthogonal Transformations.
C Int. J. Control, 33, pp. 227-246, 1981.
C
C [2] Wolovich, W.A.
C Linear Multivariable Systems, (Theorem 4.3.3).
C Springer-Verlag, 1974.
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(N ) operations.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Univ. Leuven, Belgium, March 1998.
C Supersedes Release 3.0 routine TD01OD.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Controllability, elementary polynomial operations, minimal
C realization, polynomial matrix, state-space representation,
C transfer matrix.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER ROWCOL
INTEGER INFO, LDA, LDB, LDC, LDD, LDDCOE, LDUCO1,
$ LDUCO2, LDWORK, M, NR, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INDEX(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DCOEFF(LDDCOE,*), DWORK(*),
$ UCOEFF(LDUCO1,LDUCO2,*)
C .. Local Scalars ..
LOGICAL LROCOC, LROCOR
INTEGER I, J, JSTOP, K, KDCOEF, MPLIM, MWORK, N, PWORK
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DLASET, DSWAP, TB01PD, TB01XD, TD03AY, XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX
C .. Executable Statements ..
C
INFO = 0
LROCOR = LSAME( ROWCOL, 'R' )
LROCOC = LSAME( ROWCOL, 'C' )
MPLIM = MAX( 1, M, P )
C
C Test the input scalar arguments.
C
IF( .NOT.LROCOR .AND. .NOT.LROCOC ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( ( LROCOR .AND. LDDCOE.LT.MAX( 1, P ) ) .OR.
$ ( LROCOC .AND. LDDCOE.LT.MAX( 1, M ) ) ) THEN
INFO = -6
ELSE IF( ( LROCOR .AND. LDUCO1.LT.MAX( 1, P ) ) .OR.
$ ( LROCOC .AND. LDUCO1.LT.MPLIM ) ) THEN
INFO = -8
ELSE IF( ( LROCOR .AND. LDUCO2.LT.MAX( 1, M ) ) .OR.
$ ( LROCOC .AND. LDUCO2.LT.MPLIM ) ) THEN
INFO = -9
END IF
C
N = 0
IF ( INFO.EQ.0 ) THEN
IF ( LROCOR ) THEN
C
C Initialization for T(s) given as rows over common
C denominators.
C
PWORK = P
MWORK = M
ELSE
C
C Initialization for T(s) given as columns over common
C denominators.
C
PWORK = M
MWORK = P
END IF
C
C Calculate N, the order of the resulting state-space
C representation.
C
KDCOEF = 0
C
DO 10 I = 1, PWORK
KDCOEF = MAX( KDCOEF, INDEX(I) )
N = N + INDEX(I)
10 CONTINUE
C
KDCOEF = KDCOEF + 1
C
IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDC.LT.MPLIM ) THEN
INFO = -16
ELSE IF( ( LROCOR .AND. LDD.LT.MAX( 1, P ) ) .OR.
$ ( LROCOC .AND. LDD.LT.MPLIM ) ) THEN
INFO = -18
ELSE IF( LDWORK.LT.MAX( 1, N + MAX( N, 3*M, 3*P ) ) ) THEN
INFO = -22
END IF
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'TD04AD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( MAX( N, M, P ).EQ.0 ) THEN
NR = 0
DWORK(1) = ONE
RETURN
END IF
C
IF ( LROCOC ) THEN
C
C Initialize the remainder of the leading
C MPLIM-by-MPLIM-by-KDCOEF part of U(s) to zero.
C
IF ( P.LT.M ) THEN
C
DO 20 K = 1, KDCOEF
CALL DLASET( 'Full', M-P, MPLIM, ZERO, ZERO,
$ UCOEFF(P+1,1,K), LDUCO1 )
20 CONTINUE
C
ELSE IF ( P.GT.M ) THEN
C
DO 30 K = 1, KDCOEF
CALL DLASET( 'Full', MPLIM, P-M, ZERO, ZERO,
$ UCOEFF(1,M+1,K), LDUCO1 )
30 CONTINUE
C
END IF
C
IF ( MPLIM.NE.1 ) THEN
C
C Non-scalar T(s) factorized by columns: transpose it (i.e.
C U(s)).
C
JSTOP = MPLIM - 1
C
DO 50 K = 1, KDCOEF
C
DO 40 J = 1, JSTOP
CALL DSWAP( MPLIM-J, UCOEFF(J+1,J,K), 1,
$ UCOEFF(J,J+1,K), LDUCO1 )
40 CONTINUE
C
50 CONTINUE
C
END IF
END IF
C
C Construct non-minimal state-space representation (by Wolovich's
C Structure Theorem) which has transfer matrix T(s) or T'(s) as
C appropriate ...
C
CALL TD03AY( MWORK, PWORK, INDEX, DCOEFF, LDDCOE, UCOEFF, LDUCO1,
$ LDUCO2, N, A, LDA, B, LDB, C, LDC, D, LDD, INFO )
IF ( INFO.GT.0 )
$ RETURN
C
C and then separate out a minimal realization from this.
C
C Workspace: need N + MAX(N, 3*MWORK, 3*PWORK).
C
CALL TB01PD( 'Minimal', 'Scale', N, MWORK, PWORK, A, LDA, B, LDB,
$ C, LDC, NR, TOL, IWORK, DWORK, LDWORK, INFO )
C
IF ( LROCOC ) THEN
C
C If T(s) originally factorized by columns, find dual of minimal
C state-space representation, and reorder the rows and columns
C to get an upper block Hessenberg state dynamics matrix.
C
K = IWORK(1)+IWORK(2)-1
CALL TB01XD( 'D', NR, MWORK, PWORK, K, NR-1, A, LDA, B, LDB,
$ C, LDC, D, LDD, INFO )
IF ( MPLIM.NE.1 ) THEN
C
C Also, retranspose U(s) if this is non-scalar.
C
DO 70 K = 1, KDCOEF
C
DO 60 J = 1, JSTOP
CALL DSWAP( MPLIM-J, UCOEFF(J+1,J,K), 1,
$ UCOEFF(J,J+1,K), LDUCO1 )
60 CONTINUE
C
70 CONTINUE
C
END IF
END IF
C
RETURN
C *** Last line of TD04AD ***
END