## SB08FD

### Right coprime factorization with prescribed stability degree

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

Purpose

      To construct, for a given system G = (A,B,C,D), a feedback
      matrix F and an orthogonal transformation matrix Z, such that
      the systems

           Q = (Z'*(A+B*F)*Z, Z'*B, (C+D*F)*Z, D)
      and
           R = (Z'*(A+B*F)*Z, Z'*B, F*Z, I)

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

      where G, Q and R are the corresponding transfer-function matrices.
      The resulting state dynamics matrix of the systems Q and R has
      eigenvalues lying inside a given stability domain.
      The Z matrix is not explicitly computed.

      Note: If the given state-space representation is not stabilizable,
      the unstabilizable part of the original system is automatically
      deflated and the order of the systems Q and R is accordingly
      reduced.

Specification

          SUBROUTINE SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C, LDC,
         $                   D, LDD, NQ, NR, CR, LDCR, DR, LDDR, TOL, DWORK,
         $                   LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDCR, LDD, LDDR,
         $                  LDWORK, M, N, NQ, NR, P
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), ALPHA(*), B(LDB,*), C(LDC,*),
         $                  CR(LDCR,*), 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 matrix B and
              the number of columns of the matrices C and CR.  N >= 0.

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

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

      ALPHA   (input) DOUBLE PRECISION array, dimension (2)
              ALPHA(1) contains the desired stability degree to be
              assigned for the eigenvalues of A+B*F, and ALPHA(2)
              the stability margin. The eigenvalues outside the
              ALPHA(2)-stability region will be assigned to have the
              real parts equal to ALPHA(1) < 0 and unmodified
              imaginary parts for a continuous-time system
              (DICO = 'C'), or moduli equal to 0 <= ALPHA(2) < 1
              for a discrete-time system (DICO = 'D').

      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.
              On exit, the leading NQ-by-NQ part of this array contains
              the leading NQ-by-NQ part of the matrix Z'*(A+B*F)*Z, the
              state dynamics matrix of the numerator factor Q, in a
              real Schur form. The trailing 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,M)
              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, the
              input/state matrix of the numerator factor Q. The last
              NR rows of this matrix form the input/state matrix of
              a minimal realization of the denominator factor R.

      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 (C+D*F)*Z,
              the state/output matrix of the numerator factor Q.

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

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading P-by-M part of this array must contain the
              input/output matrix. D represents also the input/output
              matrix of the numerator factor Q.

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

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

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

      CR      (output) DOUBLE PRECISION array, dimension (LDCR,N)
              The leading M-by-NQ part of this array contains the
              leading M-by-NQ part of the feedback matrix F*Z, which
              moves the eigenvalues of A lying outside the ALPHA-stable
              region to values which are on the ALPHA-stability
              boundary.  The last NR columns of this matrix form the
              state/output matrix of a minimal realization of the
              denominator factor R.

      LDCR    INTEGER
              The leading dimension of array CR.  LDCR >= MAX(1,M).

      DR      (output) DOUBLE PRECISION array, dimension (LDDR,M)
              The leading M-by-M part of this array contains an
              identity matrix representing the input/output matrix
              of the denominator factor R.

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

Tolerances

      TOL     DOUBLE PRECISION
              The absolute tolerance level below which the elements of
              B are considered zero (used for controllability tests).
              If the user sets TOL <= 0, then an implicitly computed,
              default tolerance, defined by  TOLDEF = N*EPS*NORM(B),
              is used instead, where EPS is the machine precision
              (see LAPACK Library routine DLAMCH) and NORM(B) denotes
              the 1-norm of B.

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.
              LWORK >= MAX( 1, N*(N+5), 5*M, 4*P ).
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = K:  K violations of the numerical stability condition
                    NORM(F) <= 10*NORM(A)/NORM(B) 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 + B*F)*Z
                    along the diagonal.

Method

      The subroutine is based on the factorization algorithm of [1].

References

      [1] Varga A.
          Coprime factors model reduction method based on
          square-root balancing-free techniques.
          System Analysis, Modelling and Simulation,
          vol. 11, pp. 303-311, 1993.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

    *     SB08FD 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, LDCR, LDD, LDDR
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDCR = MMAX, LDD = PMAX, LDDR = MMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( NMAX*( NMAX + 5 ), 5*MMAX,
         $                                 4*PMAX ) )
    *     .. 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), ALPHA(2), B(LDB,MMAX), C(LDC,NMAX),
         $                 CR(LDCR,NMAX), D(LDD,MMAX), DR(LDDR,MMAX),
         $                 DWORK(LDWORK)
    *     .. External Subroutines ..
          EXTERNAL         SB08FD
    *     .. 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, ALPHA(1), TOL, DICO
          ALPHA(2) = ALPHA(1)
          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 RCF for (A,B,C,D).
                   CALL SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C,
         $                      LDC, D, LDD, NQ, NR, CR, LDCR, 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 = NQ-NR+1, NQ
                         WRITE ( NOUT, FMT = 99995 )
         $                     ( A(I,J), J = NQ-NR+1, NQ )
       80             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
                      DO 90 I = NQ-NR+1, NQ
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
       90             CONTINUE
                      IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
                      DO 100 I = 1, M
                         WRITE ( NOUT, FMT = 99995 )
         $                     ( CR(I,J), J = NQ-NR+1, NQ )
      100             CONTINUE
                      WRITE ( NOUT, FMT = 99983 )
                      DO 110 I = 1, M
                         WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, M )
      110             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB08FD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB08FD = ',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

     SB08FD EXAMPLE PROGRAM DATA (Continuous system)
      7  2  3 -1.0   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

     SB08FD EXAMPLE PROGRAM RESULTS

     The numerator state dynamics matrix AQ is
      -1.4178  -5.1682   3.2450  -0.2173   0.0564  -3.2129  -3.6183
       0.9109  -1.4178  -2.1262   0.1231   0.0805  -0.4392  -0.2528
       0.0000   0.0000 -13.1627   0.0608  -0.0218   2.3461   5.8272
       0.0000   0.0000   0.0000  -3.5957  -3.3373   1.3622  -3.6083
       0.0000   0.0000   0.0000   0.0000 -12.4245  -9.8634   8.1191
       0.0000   0.0000   0.0000   0.0000   0.0000  -1.0000  -0.0135
       0.0000   0.0000   0.0000   0.0000   0.0000   1.7393  -1.0000

     The numerator input/state matrix BQ is
       5.0302  -0.0063
       0.7078  -0.0409
     -11.3663   0.0051
       0.1760   0.5879
      -0.0265  12.2119
       1.0104   1.3262
       0.4474  -2.2388

     The numerator state/output matrix CQ is
      -0.8659   0.2787  -0.3432   0.0020   0.0000   0.2026   0.1172
       0.0797  -0.3951   0.0976  -0.0292   0.0062   0.7676   0.4879
      -0.0165  -0.0645   0.0097   0.8032  -0.1602   0.3050  -0.4812

     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
      -1.0000  -0.0135
       1.7393  -1.0000

     The denominator input/state matrix BR is
       1.0104   1.3262
       0.4474  -2.2388

     The denominator state/output matrix CR is
      -0.1091  -0.4653
      -0.7055   0.4766

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

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