## SB08CD

### Left coprime factorization with inner denominator

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To construct, for a given system G = (A,B,C,D), an output
      injection matrix H, an orthogonal transformation matrix Z, and a
      gain matrix V, such that the systems

           Q = (Z'*(A+H*C)*Z, Z'*(B+H*D), V*C*Z, V*D)
      and
           R = (Z'*(A+H*C)*Z, Z'*H, V*C*Z, V)

      provide a stable left coprime factorization of G in the form
                    -1
               G = R  * Q,

      where G, Q and R are the corresponding transfer-function matrices
      and the denominator R is co-inner, that is, R(s)*R'(-s) = I in
      the continuous-time case, or R(z)*R'(1/z) = I in the discrete-time
      case. The Z matrix is not explicitly computed.

      Note: G must have no observable poles on the imaginary axis
      for a continuous-time system, or on the unit circle for a
      discrete-time system. If the given state-space representation
      is not detectable, the undetectable part of the original
      system is automatically deflated and the order of the systems
      Q and R is accordingly reduced.

Specification

          SUBROUTINE SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   NQ, NR, BR, LDBR, DR, LDDR, TOL, DWORK, LDWORK,
         $                   IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO
          INTEGER           INFO, IWARN, LDA, LDB, LDBR, LDC, LDD, LDDR,
         $                  LDWORK, M, N, NQ, NR, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), BR(LDBR,*), C(LDC,*),
         $                  D(LDD,*), DR(LDDR,*), DWORK(*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the original system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

Input/Output Parameters

      N       (input) INTEGER
              The dimension of the state vector, i.e. the order of the
              matrix A, and also the number of rows of the matrices B
              and BR, and the number of columns of the matrix C.
              N >= 0.

      M       (input) INTEGER
              The dimension of input vector, i.e. the number of columns
              of the matrices B and D.  M >= 0.

      P       (input) INTEGER
              The dimension of output vector, i.e. the number of rows
              of the matrices C, D and DR, and the number of columns
              of the matrices BR and DR.  P >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the state dynamics matrix A. The matrix A must not
              have observable eigenvalues on the imaginary axis, if
              DICO = 'C', or on the unit circle, if DICO = 'D'.
              On exit, the leading NQ-by-NQ part of this array contains
              the leading NQ-by-NQ part of the matrix Z'*(A+H*C)*Z, the
              state dynamics matrix of the numerator factor Q, in a
              real Schur form. The leading NR-by-NR part of this matrix
              represents the state dynamics matrix of a minimal
              realization of the denominator factor R.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension
              (LDB,MAX(M,P))
              On entry, the leading N-by-M part of this array must
              contain the input/state matrix.
              On exit, the leading NQ-by-M part of this array contains
              the leading NQ-by-M part of the matrix Z'*(B+H*D), the
              input/state matrix of the numerator factor Q.
              The remaining part of this array is needed as workspace.

      LDB     INTEGER
              The leading dimension of array B.  LDB >= MAX(1,N).

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the state/output matrix C.
              On exit, the leading P-by-NQ part of this array contains
              the leading P-by-NQ part of the matrix V*C*Z, the
              state/output matrix of the numerator factor Q.
              The first NR columns of this array represent the
              state/output matrix of a minimal realization of the
              denominator factor R.
              The remaining part of this array is needed as workspace.

      LDC     INTEGER
              The leading dimension of array C.
              LDC >= MAX(1,M,P), if N > 0.
              LDC >= 1,          if N = 0.

      D       (input/output) DOUBLE PRECISION array, dimension
              (LDD,MAX(M,P))
              On entry, the leading P-by-M part of this array must
              contain the input/output matrix.
              On exit, the leading P-by-M part of this array contains
              the matrix V*D representing the input/output matrix
              of the numerator factor Q.
              The remaining part of this array is needed as workspace.

      LDD     INTEGER
              The leading dimension of array D.  LDD >= MAX(1,M,P).

      NQ      (output) INTEGER
              The order of the resulting factors Q and R.
              Generally, NQ = N - NS, where NS is the number of
              unobservable eigenvalues outside the stability region.

      NR      (output) INTEGER
              The order of the minimal realization of the factor R.
              Generally, NR is the number of observable eigenvalues
              of A outside the stability region (the number of modified
              eigenvalues).

      BR      (output) DOUBLE PRECISION array, dimension (LDBR,P)
              The leading NQ-by-P part of this array contains the
              leading NQ-by-P part of the output injection matrix
              Z'*H, which reflects the eigenvalues of A lying outside
              the stable region to values which are symmetric with
              respect to the imaginary axis (if DICO = 'C') or the unit
              circle (if DICO = 'D'). The first NR rows of this matrix
              form the input/state matrix of a minimal realization of
              the denominator factor R.

      LDBR    INTEGER
              The leading dimension of array BR.  LDBR >= MAX(1,N).

      DR      (output) DOUBLE PRECISION array, dimension (LDDR,P)
              The leading P-by-P part of this array contains the lower
              triangular matrix V representing the input/output matrix
              of the denominator factor R.

      LDDR    INTEGER
              The leading dimension of array DR.  LDDR >= MAX(1,P).

Tolerances

      TOL     DOUBLE PRECISION
              The absolute tolerance level below which the elements of
              C are considered zero (used for observability tests).
              If the user sets TOL <= 0, then an implicitly computed,
              default tolerance, defined by  TOLDEF = N*EPS*NORM(C),
              is used instead, where EPS is the machine precision
              (see LAPACK Library routine DLAMCH) and NORM(C) denotes
              the infinity-norm of C.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of working array DWORK.
              LDWORK >= MAX( 1, P*N + MAX( N*(N+5),P*(P+2),4*P,4*M ) ).
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = K:  K violations of the numerical stability condition
                    NORM(H) <= 10*NORM(A)/NORM(C) occured during the
                    assignment of eigenvalues.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the reduction of A to a real Schur form failed;
              = 2:  a failure was detected during the ordering of the
                    real Schur form of A, or in the iterative process
                    for reordering the eigenvalues of Z'*(A + H*C)*Z
                    along the diagonal;
              = 3:  if DICO = 'C' and the matrix A has an observable
                    eigenvalue on the imaginary axis, or DICO = 'D' and
                    A has an observable eigenvalue on the unit circle.

Method

      The subroutine uses the right coprime factorization algorithm with
      inner denominator of [1] applied to G'.

References

      [1] Varga A.
          A Schur method for computing coprime factorizations with
          inner denominators and applications in model reduction.
          Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.

Numerical Aspects

                                             3
      The algorithm requires no more than 14N  floating point
      operations.

Further Comments

      None

Example

Program Text

    *     SB08CD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          MPMAX
          PARAMETER        ( MPMAX = MAX( MMAX, PMAX ) )
          INTEGER          LDA, LDB, LDBR, LDC, LDD, LDDR
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDBR = NMAX,
         $                   LDC = MPMAX, LDD = MPMAX, LDDR = PMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX*PMAX + MAX( NMAX*( NMAX + 5 ),
         $                            PMAX*( PMAX + 2 ), 4*PMAX, 4*MMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, IWARN, J, M, N, NQ, NR, P
          CHARACTER*1      DICO
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MPMAX), BR(LDBR,PMAX),
         $                 C(LDC,NMAX), D(LDD,MPMAX), DR(LDDR,PMAX),
         $                 DWORK(LDWORK)
    *     .. External Subroutines ..
          EXTERNAL         SB08CD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, P, TOL, DICO
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1, N ), I = 1, N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99989 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1, M ), I = 1, N )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99988 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1, N ), I = 1, P )
                   READ ( NIN, FMT = * ) ( ( D(I,J), J = 1, M ), I = 1, P )
    *              Find a RCFID for (A,B,C,D).
                   CALL SB08CD( DICO, N, M, P, A, LDA, B, LDB, C, LDC,
         $                      D, LDD, NQ, NR, BR, LDBR, DR, LDDR, TOL,
         $                      DWORK, LDWORK, IWARN, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, NQ
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
       20             CONTINUE
                      IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, NQ
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
       40             CONTINUE
                      IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
       70             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
                      DO 80 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 )
         $                     ( A(I,J), J = 1, NR )
       80             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
                      DO 90 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( BR(I,J), J = 1, P )
       90             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
                      DO 100 I = 1, P
                         WRITE ( NOUT, FMT = 99995 )
         $                     ( C(I,J), J = 1, NR )
      100             CONTINUE
                      WRITE ( NOUT, FMT = 99983 )
                      DO 110 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, P )
      110             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB08CD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB08CD = ',I2)
    99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
    99995 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' The numerator input/state matrix BQ is ')
    99992 FORMAT (/' The numerator state/output matrix CQ is ')
    99991 FORMAT (/' The numerator input/output matrix DQ is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' P is out of range.',/' P = ',I5)
    99986 FORMAT (/' The denominator state dynamics matrix AR is ')
    99985 FORMAT (/' The denominator input/state matrix BR is ')
    99984 FORMAT (/' The denominator state/output matrix CR is ')
    99983 FORMAT (/' The denominator input/output matrix DR is ')
          END

Program Data

     SB08CD EXAMPLE PROGRAM DATA (Continuous system)
      7  2  3   1.E-10 C
     -0.04165  0.0000  4.9200   0.4920  0.0000   0.0000  0.0000
     -5.2100  -12.500  0.0000   0.0000  0.0000   0.0000  0.0000
      0.0000   3.3300 -3.3300   0.0000  0.0000   0.0000  0.0000
      0.5450   0.0000  0.0000   0.0000  0.0545   0.0000  0.0000
      0.0000   0.0000  0.0000  -0.49200 0.004165 0.0000  4.9200
      0.0000   0.0000  0.0000   0.0000  0.5210  -12.500  0.0000
      0.0000   0.0000  0.0000   0.0000  0.0000   3.3300 -3.3300
      0.0000   0.0000
      12.500   0.0000
      0.0000   0.0000
      0.0000   0.0000
      0.0000   0.0000
      0.0000   12.500
      0.0000   0.0000
      1.0000   0.0000  0.0000   0.0000  0.0000  0.0000  0.0000
      0.0000   0.0000  0.0000   1.0000  0.0000  0.0000  0.0000
      0.0000   0.0000  0.0000   0.0000  1.0000  0.0000  0.0000
      0.0000   0.0000
      0.0000   0.0000
      0.0000   0.0000

Program Results

     SB08CD EXAMPLE PROGRAM RESULTS

     The numerator state dynamics matrix AQ is
      -0.1605   0.0523   0.9423   2.0193   0.4166   0.2518   1.6140
      -0.4489  -0.1605   1.7955   3.8719  -0.2394   0.0491  -0.8740
       0.0000   0.0000 -12.4245   3.5463  -0.0057   0.0254  -0.0053
       0.0000   0.0000   0.0000  -3.5957  -0.0153  -0.0290  -0.0616
       0.0000   0.0000   0.0000   0.0000 -13.1627  -1.9835  -3.6182
       0.0000   0.0000   0.0000   0.0000   0.0000  -1.4178   5.6218
       0.0000   0.0000   0.0000   0.0000   0.0000  -0.8374  -1.4178

     The numerator input/state matrix BQ is
      -1.0157   0.2554
       0.5523   0.4443
       0.0056 -11.6989
       0.0490   4.3728
      11.7198  -0.0038
      -2.8173   0.0308
       3.1018  -0.0009

     The numerator state/output matrix CQ is
       0.1975  -0.1063  -0.0006  -0.0083   0.1279   0.8797   0.3994
       0.8541  -0.4513  -0.0007  -0.0041   0.0305  -0.2562   0.0122
       0.4668   0.8826   0.0248  -0.0506   0.0000   0.0022  -0.0017

     The numerator input/output matrix DQ is
       0.0000   0.0000
       0.0000   0.0000
       0.0000   0.0000

     The denominator state dynamics matrix AR is
      -0.1605   0.0523
      -0.4489  -0.1605

     The denominator input/state matrix BR is
      -0.0158  -0.0692  -0.1688
       0.0306   0.1281  -0.4984

     The denominator state/output matrix CR is
       0.1975  -0.1063
       0.8541  -0.4513
       0.4668   0.8826

     The denominator input/output matrix DR is
       1.0000   0.0000   0.0000
       0.0000   1.0000   0.0000
       0.0000   0.0000   1.0000

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