## AB09DD

### Computation of a reduced order model using singular perturbation approximation formulas

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

Purpose

      To compute a reduced order model by using singular perturbation
      approximation formulas.

Specification

          SUBROUTINE AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
         $                   D, LDD, RCOND, IWORK, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO
          INTEGER           INFO, LDA, LDB, LDC, LDD, M, N, NR, P
          DOUBLE PRECISION  RCOND
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), DWORK(*)
          INTEGER           IWORK(*)

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; also the number of rows of matrix B 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 matrices B and D.  M >= 0.

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

      NR      (input) INTEGER
              The order of the reduced order system.  N >= NR >= 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 of the original system.
              On exit, the leading NR-by-NR part of this array contains
              the state dynamics matrix Ar of the reduced order system.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input/state matrix of the original system.
              On exit, the leading NR-by-M part of this array contains
              the input/state matrix Br of the reduced order system.

      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 of the original system.
              On exit, the leading P-by-NR part of this array contains
              the state/output matrix Cr of the reduced order system.

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

      D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
              On entry, the leading P-by-M part of this array must
              contain the input/output matrix of the original system.
              On exit, the leading P-by-M part of this array contains
              the input/output matrix Dr of the reduced order system.
              If NR = 0 and the given system is stable, then D contains
              the steady state gain of the system.

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

      RCOND   (output) DOUBLE PRECISION
              The reciprocal condition number of the matrix A22-g*I
              (see METHOD).

Workspace

      IWORK   INTEGER array, dimension (2*(N-NR))

      DWORK   DOUBLE PRECISION array, dimension (4*(N-NR))

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1: if the matrix A22-g*I (see METHOD) is numerically
                   singular.

Method

      Given the system (A,B,C,D), partition the system matrices as

             ( A11 A12 )        ( B1 )
         A = (         ) ,  B = (    ) ,  C = ( C1  C2 ),
             ( A21 A22 )        ( B2 )

      where A11 is NR-by-NR, B1 is NR-by-M, C1 is P-by-NR, and the other
      submatrices have appropriate dimensions.

      The matrices of the reduced order system (Ar,Br,Cr,Dr) are
      computed according to the following residualization formulas:
                                 -1                               -1
         Ar = A11 + A12*(g*I-A22)  *A21 ,  Br = B1 + A12*(g*I-A22)  *B2
                               -1                               -1
         Cr = C1 + C2*(g*I-A22)  *A21   ,  Dr = D + C2*(g*I-A22)  *B2

      where g = 0 if DICO = 'C' and g = 1 if DICO = 'D'.

Further Comments

      None

Example

Program Text

    *     AB09DD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDD
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = 2*NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 4*NMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION RCOND
          INTEGER          I, INFO, J, M, N, NR, P
          CHARACTER*1      DICO
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DWORK(LDWORK)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         AB09DD
    *     .. 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, NR, 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 reduced ssr for (A,B,C).
                   CALL AB09DD( DICO, N, M, P, NR, A, LDA, B, LDB, C, LDC,
         $                      D, LDD, RCOND, IWORK, DWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) RCOND
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
       20             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, NR
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB09DD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB09DD = ',I2)
    99997 FORMAT (' The computed reciprocal condition number = ',1PD12.5)
    99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
    99995 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' The reduced input/state matrix Br is ')
    99992 FORMAT (/' The reduced state/output matrix Cr is ')
    99991 FORMAT (/' The reduced input/output matrix Dr 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)
          END

Program Data

     AB09DD EXAMPLE PROGRAM DATA  (Continuous system)
      7     2     3     5      C
      -0.04165    4.9200   -4.9200         0         0         0         0
             0   -3.3300         0         0         0    3.3300         0
        0.5450         0         0   -0.5450         0         0         0
             0         0    4.9200  -0.04165    4.9200         0         0
             0         0         0         0   -3.3300         0    3.3300
       -5.2100         0         0         0         0  -12.5000         0
             0         0         0   -5.2100         0         0  -12.5000
             0         0
             0         0
             0         0
             0         0
             0         0
       12.5000         0
             0   12.5000
         1     0     0     0     0     0     0
         0     0     1     0     0     0     0
         0     0     0     1     0     0     0
         0     0
         0     0
         0     0

Program Results

     AB09DD EXAMPLE PROGRAM RESULTS

     The computed reciprocal condition number =  1.00000D+00

     The reduced state dynamics matrix Ar is
      -0.0416   4.9200  -4.9200   0.0000   0.0000
      -1.3879  -3.3300   0.0000   0.0000   0.0000
       0.5450   0.0000   0.0000  -0.5450   0.0000
       0.0000   0.0000   4.9200  -0.0416   4.9200
       0.0000   0.0000   0.0000  -1.3879  -3.3300

     The reduced input/state matrix Br is
       0.0000   0.0000
       3.3300   0.0000
       0.0000   0.0000
       0.0000   0.0000
       0.0000   3.3300

     The reduced state/output matrix Cr is
       1.0000   0.0000   0.0000   0.0000   0.0000
       0.0000   0.0000   1.0000   0.0000   0.0000
       0.0000   0.0000   0.0000   1.0000   0.0000

     The reduced input/output matrix Dr is
       0.0000   0.0000
       0.0000   0.0000
       0.0000   0.0000

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